aa r X i v : . [ m a t h . C O ] O c t Counting packings of generic subsets in finite groups
Roland BacherAugust 24, 2018
Abstract : A packing of subsets S , . . . , S n in a group G is an element ( g , . . . , g n ) of G n such that g S , . . . , g n S n are disjoint subsets of G . Wegive a formula for the number of packings if the group G is finite and ifthe subsets S , . . . , S n satisfy a genericity condition. This formula can beseen as a generalization of the falling factorials which encode the number ofpackings in the case where all the sets S i are singletons. A (left-)packing of n non-empty subsets S , . . . , S n in a group G is an element( g , . . . , g n ) of G n such that the left-translates g S , . . . , g n S n of the sets S i are disjoint. The sets S , . . . , S n are labelled by their indices. In particular,permuting the elements g , . . . , g n of a packing ( g , . . . , g n ) ∈ G n of S = · · · = S n yields a different packing. Moreover, in the case where S forexample is of the form S = H S for some subgroup H of G , a packing( g , . . . , g n ) gives rise to ♯ ( H ) distinct packings ( g h, g , . . . , g n ) , h ∈ H .There is an obvious one-to-one map between packings of S , . . . , S n ⊂ G and packings of a S , . . . , a n S n ⊂ G for every ( a , . . . , a n ) ∈ G n .This paper deals with enumerative properties of left-packings in the casewhere G is a finite group. Using the involutive antiautomorphism g g − , its content can easily be modified in order to deal with right-packings S g , . . . , S n g n .In the sequel, we denote by α ( G ; S , . . . , S n ) ≤ N n the number of pack-ings of n non-empty subsets S , . . . , S n in a finite group G with N elements.Computing α ( G ; S , . . . , S n ) for arbitrary subsets S , . . . , S n in a finite group G is probably difficult. There are however easy lower and upper bounds: Proposition 1.1.
We set a = α ( G ; S , . . . , S n ) and b = α ( G ; S , . . . , S n , S n +1 ) where S , . . . , S n +1 are ( n + 1) non-empty subsets in a finite group G . We Keywords: Enumerative combinatorics, packings in groups, additive combinatorics,additive number theory, Stirling number. Math. class: 05A15, 05C30, 11B73, 11P99 ave the inequalities N − ♯ ( S n +1 ) n X i =1 ♯ ( S i ) ! a ≤ b ≤ N − n X i =1 ♯ ( S i ) ! a . In particular, we have b = N − n X i =1 ♯ ( S i ) ! a (1) if S n +1 is a singleton. Proposition 1.1 will be proven in Section 3.A family S , . . . , S n of n non-empty subsets in a group G with identityelement e is generic if for every sequence i , . . . , i k of k distinct elements in { , . . . , n } and for every choice of elements g i j ∈ S − i j S i j \ { e } , we have g i g i · · · g i k = e . Otherwise stated, a family S , . . . , S n of subsets in a group G is genericif the only solution of the equations g i · · · g i n = e with g i j ∈ S − i j S i j for { i , . . . , i n } = { , . . . , n } is given by g i j = e for all j .Genericity excludes “accidental intersections”among translates g S , . . . , g n S n in the following sense: Given a collection of translates g S , . . . , g n S n , weconsider the associated intersection graph with vertices S i and edges joining S i , S j if g i S i ∩ g j S j = ∅ . Genericity of a family S , . . . , S j in a group G isequivalent to the statement that all intersection graphs are primal graphsof hyperforests. Intuitively speaking, intersections among translates of ageneric family are always “as small as possible”. Example.
Genericity in an additive abelian group G boils down to thefact that the subset ( S − S ) × · · · × ( S n − S n ) of the group G n intersectsthe subgroup { ( x , . . . , x n ) ∈ G n | P ni =1 x i = 0 } of G n only in the identityelement (0 , . . . , S , . . . , S n of subsets in the additive group Z with pre-scribed cardinalities s i = ♯ ( S i ) can be constructed by starting with S = { , . . . , s − } and by defining S i recursively as S i = { , k i , k i , . . . , ( s i − k i } where k i is an arbitrary natural integer strictly larger than P i − j =1 (max( S j ) − min( S j )) = P i − j =1 ( s j − k j . A generic family is thus for example given by the sets S = { , } , S = { , } , . . . , S i = { , i − } , . . . , S n = { , n − } .Reduction of a generic family S , . . . , S n ⊂ Z modulo a natural integer N yields a generic family of Z /N Z except if N is a divisor of a non-zerointeger in the finite set { P ni =1 ( S i − S i ) } . Remark 1.2.
The terminology “generic family” can be motivated as fol-lows: Given n strictly positive natural numbers s , . . . , s n , most uniform andom choices of n subsets S , . . . , S n with ♯ ( S i ) = s i (among all (cid:0) Ns i (cid:1) pos-sible subsets) in a finite group G of order N should yield a generic familyif N is large compared to P nk =2 k ! τ k with τ , . . . , τ n defined by P nk =0 τ k t k = Q nj =1 (1 + s j ( s j − t ) . Indeed, the number k ! τ k is an upper bound on thenumber of elements in the set E k containing all products of the form g i · · · g i k with g i j ∈ S − i j S i j \ { e } and i , . . . , i k given by k ≥ distinct elements of { , . . . , n } . Under the (naive but hopefully correct) assumption that the ele-ments of E k are uniformly distributed in G , the probability for non-genericityof S , . . . , S n is at most N P nk =2 k ! κ k . Observe also the trivial inequalities P nk =2 k ! κ k < n ! P nk =0 κ k = n ! Q nj =1 (1 + s j ( s j − ≤ n ! Q nj =1 s j . The aim of this paper is to describe a universal formula for the num-ber of packings for a generic family of subsets S , . . . , S n in a finite group G .The number of associated packings depends then only on the cardinalities of G and S , . . . , S n . Moreover for fixed cardinalities of S , . . . , S n , the depen-dency on the cardinality of G is polynomial of degree n . A trivial example isthe generic family given by n subsets reduced to singletons. The associatednumber of packings in a finite group with N elements is then easily seen tobe given by the polynomial n ! (cid:0) Nn (cid:1) = N ( N − · · · ( N − n + 1) ∈ Z [ N ] withcoefficients given by Stirling numbers of the first kind. This polynomial isalso called a falling factorial and denoted by N n . Using the formulae ofour paper, it is possible to define the falling factorial N λ associated to apartition λ = λ , λ , . . . by counting packings of generic families with λ subsets having ν , ν , . . . , ν λ elements where ν i = { j | λ j ≥ i } is the i − thpart of the transposed partition ν = λ t of λ . The map λ N λ is how-ever perhaps not exceedingly interesting. On one hand, it is not into since N λ = N for every partition λ of the form 1 , , , . . . . On the other hand,fixing the content P j λ j of the partition λ , our formulae show that thecoefficients of N λ depend linearly on the elementary symmetric functions σ = P i 1) + t i − ,j − ( n − 1) + ( i − t i − ,j ( n − 1) (2)for n ≥ 2. We set t i,j ( n ) = 0 in all other cases, i.e. if i ≤ n or j < i + j > n .Given a natural integer n ≥ 1, the set of all (cid:0) n +12 (cid:1) non-zero integers t i,j ( n ) can be organized into a triangular array T ( n ) with rows indexedby { n + 1 , . . . , n } and columns indexed by { , . . . , n − } such that T ( n )determines T ( n +1) recursively by Formula (2) reminiscent of the recurrencerelation (cid:0) nk (cid:1) = (cid:0) n − k − (cid:1) + (cid:0) n − k (cid:1) for binomial coefficients. The first six triangulararrays T (1) , . . . , T (6) are1 1 11 2 3 15 33 6 11 6 126 26 635 151524 50 35 10 1154 200 80 10340 255 45315 105105 120 274 225 85 15 11044 1604 855 190 153304 3325 1050 1054900 2940 4203465 945945Observe that the first row of T (1) , T (2) , . . . coincides, up to signs, withStirling numbers of the first kind. More precisely, we have n − X k =0 t n +1 ,k ( n ) x k +1 = n − Y j =0 ( x + j ) = ( − n n X j =1 S ( n, j )( − x ) j . (3)This is of course an easy consequence of the recurrence relation (2). Theintegers t i,j ( n ) seem to be related to a few interesting integer-sequences, seeSection 11 for examples.We consider the formal power series U ∈ A [[ x ]] with coefficients in thering A = Z [ σ , σ , σ , . . . ] of integral polynomials in σ , σ , . . . defined by U ( x, σ , σ , . . . ) = 1 − ∞ X n =1 x n n X i = n +1 σ i n − i X j =0 t i,j ( n )( − σ ) j . (4) Theorem 2.1. The number of packings of a generic family S , . . . , S n of n non-empty subsets in a finite group G with N elements equals N n U ( N − , σ , σ , . . . ) (5)5 or U given by Formula (4) and for σ , σ , . . . defined by ∞ X j =0 σ j t j = n Y k =1 (1 + ♯ ( S k ) t ) . Remark that Formula (5) of Theorem 2.1 is polynomial of degree n in N for fixed complex numbers σ , σ , . . . such that σ n +1 = σ n +2 = · · · =0. Indeed, the coefficient of x m in U ( x, σ , σ , . . . ) belongs to the idealgenerated by σ m +1 , σ m +2 , . . . , σ m of Z [ σ , σ , . . . ] and is thus zero for m ≥ n if σ n +1 = σ n +2 = · · · = 0.The ingredients for proving Theorem 2.1 are the following four results: Proposition 2.2. There exists a series U ∈ Z [[ x, σ , σ , . . . ]] such thatFormula (5) with σ , σ , . . . defined as in Theorem 2.1 gives the number ofpackings for every generic family of n non-empty subsets in a finite groupwith N elements.Moreover, the coefficient of a non-constant monomial x m in this series U is of degree at most m with respect to the grading deg σ i = i and belongsto the ideal of Z [ σ , σ , . . . ] generated by σ m +1 , σ m +2 , . . . , σ m . The proof of Proposition 2.2 relies on combinatorial properties of inter-section graphs encoding non-trivial intersections among subsets g S , . . . , g n S n of a group G . These properties are encoded by the poset HF ( n ) of hyper-forests with n labelled vertices and order relation given by F ′ ≤ F if everyhyperedge of F ′ is contained in some hyperedge of F . The poset HF ( n ) is alattice with minimal element the trivial graph defined by n isolated labelledvertices and with maximal element the hypertree consisting of a unique hy-peredge containing all n labelled vertices. Our proof of Proposition 2.2 usesM¨obius inversion in H F ( n ). It needs only the existence (which is obvious)of a M¨obius function on the poset HF ( n ). The explicit description of U given by Theorem 2.1 allows however a posteriori the computation (givenby Proposition 7.1) of the M¨obius function of HF ( n ). Remark that theposet HT n of hypertrees with n labelled vertices appearing for example in[3] is a subposet of the order dual of HF ( n ) obtained by restricting theinverse order of HF ( n ) to the subset of all hypertrees in HF ( n ). Proposition 2.3. A series U as in Proposition 2.2 satisfies the functionalequation (1 − σ x ) U ( x, σ , σ , σ , . . . ) = U ( x, ˜ σ , ˜ σ , ˜ σ , . . . ) (6) where ˜ σ i = σ i − + σ i , using the convention σ = 1 . Proof Equation (6) corresponds to equation (1) if σ , σ , . . . are elemen-tary symmetric functions of a finite set of natural integers. The general casefollows by remarking that the algebra of symmetric polynomials is a freepolynomial algebra on the set of elementary symmetric polynomials. ✷ roposition 2.4. The series U defined by Formula (4) satisfies the func-tional equation (6). Proposition 2.5. The functional equation (6) has at most one solution ofthe form U = 1 + . . . such that the coefficient of a nonconstant monomial x n is of degree at most n (with respect to the grading deg σ i = i ) and belongsto the ideal generated by σ n +1 , σ n +2 , . . . , σ n in Z [ σ , σ , . . . ] . Proof of Theorem 2.1 Proposition 2.2 ensures the existence of a seriesenumerating packings of generic families in finite groups. This series coin-cides with the series given by Formula (4) by Propositions 2.3, 2.4 and 2.5. ✷ Remark 2.6. Iterating identity (6) n times we have U ( x, σ , σ , . . . ) n − Y j =0 (1 − ( σ + j ) x ) = U ( x, ˜ σ , ˜ σ , ˜ σ , . . . ) where ˜ σ k = min( k,n ) X j =0 (cid:18) nj (cid:19) σ k − j . A particular case is the specialization U (cid:18) x, (cid:18) n (cid:19) , (cid:18) n (cid:19) , (cid:18) n (cid:19) , . . . (cid:19) = n − Y j =1 (1 − jx ) associated to generic families S , . . . , S n given by n singletons. Remark 2.7. It is widespread lore that interesting combinatorial identitieshave q − analogues generally encoding an additional feature of the involvedcombinatorial objects. I do not know if the integers t i,j ( n ) or the series U have such a q − analogue with interesting properties. S , . . . , S n of subsets in a group G A packing of S , . . . , S n given by ( g , . . . , g n ) ∈ G n extends to a pack-ing ( g , . . . , g n , g n +1 ) ∈ G n +1 of S , . . . , S n +1 if and only if g n +1 ∈ G \ (cid:0) ∪ ni =1 g i S i ( S n +1 ) − (cid:1) where S − = { g − | g ∈ S} . Since g i S i ( S n +1 ) − con-tains at most ♯ ( S n +1 ) ♯ ( S i ) elements, we have the first inequality.Considering a fixed element h ∈ S n +1 we have the inequality ♯ (cid:0) ∪ ni =1 g i S i ( S n +1 ) − (cid:1) ≥ ♯ (cid:0) ∪ ni =1 g i S i h − (cid:1) = ♯ ( ∪ ni =1 g i S i ) . g , . . . , g n ), we have ♯ ( ∪ ni =1 g i S i ) = n X i =1 ♯ ( S i )showing the second inequality.Both inequalities are sharp if ♯ ( S n +1 ) = 1. This proves equality (1). ✷ We fix a group G and a family S , . . . , S n of n non-empty subsets in G .Given an element g = ( g , . . . , g n ) of G n , we consider the corresponding intersection graph I ( g ) with vertices 1 , . . . , n and edges { i, j } between dis-tinct vertices i, j if g i S i ∩ g j S j = ∅ in G . Observe that g = ( g , . . . , g n ) in G n defines a packing if and only if I ( g ) is the trivial graph with n isolatedvertices.Given a finite simple graph Γ with vertices 1 , . . . , n and edges E (Γ), weconsider the set R Γ = { ( g , . . . , g n ) ∈ G n | g i S i ∩ g j S j = ∅ for every { i, j } ∈ E (Γ) } . An element g in G n belongs thus to R Γ if and only if Γ is a subgraph ofthe intersection graph I ( g ).We denote by E Γ the set of equivalence classes of R Γ defined by ( g , . . . , g n ) ∼ ( h , . . . , h n ) if g i h − i = g j h − j for every edge { i, j } of Γ. Two elements g = ( g , . . . , g n ) and h = ( h , . . . , h n ) of R Γ represent thus the same equiv-alence class of E Γ if and only if the map i g i h − i is constant on (verticesof) connected components. Proposition 3.1. Suppose that G is a finite group with N elements. Wehave then ♯ ( R Γ ) = ♯ ( E Γ ) N c (Γ) where c (Γ) denotes the number of connected components of Γ . Proof We set c = c (Γ) and we denote the connected components of Γby Γ , . . . , Γ c . We get a free action of G c on R Γ by considering( a , . . . , a c ) · ( g , . . . , g n ) ( a − γ (1) g , . . . , a − γ ( n ) g n )where γ ( i ) ∈ { , . . . , c } is defined by the inclusion of the vertex i in the γ ( i ) − th connected component Γ γ ( i ) of Γ. Orbits in R Γ of this action arethus in one-to-one correspondence with equivalence classes of E Γ . ✷ Remark 3.2. The set E Γ associated to a graph Γ with c connected compo-nents contains at most (max i ♯ ( S i )) n − c distinct equivalence classes. Indeed,we have R (Γ ′ ) ⊂ R (Γ) if Γ is a subgraph of Γ ′ . Replacing Γ by a spanning orest, we can thus assume that Γ is a forest. The equivalence class of anelement g ∈ R (Γ) is now determined by the relative positions of g i S i and g j S j for all n − c edges { i, j } of the forest Γ and the number of differentrelative positions of g i S i and g j S j is at most ♯ ( S i ) ♯ ( S j ) ≤ (max i ♯ ( S i )) . Proposition 3.3. The number α = α ( G ; S , . . . , S n ) of packings of a family S , . . . , S n in a finite group G with N elements is given by α = X Γ ∈B ( − e (Γ) ♯ ( E Γ ) N c (Γ) where the sum is over the Boolean poset B of all n ) simple graphs withvertices , . . . , n and where e (Γ) = ♯ ( E (Γ)) , respectively c (Γ) , denotes thenumber of edges, respectively connected components, of a graph Γ ∈ B . Proof Proposition 3.1 shows that it is enough to prove the equality α = X Γ ∈B ( − e (Γ) ♯ ( R Γ ) . An element g = ( g , . . . , g n ) ∈ G n defines a packing if and only if its inter-section graph I ( g ) is trivial. It provides thus a contribution of 1 to α in thiscase since it is only involved as an element of R Γ if Γ is the trivial graphwith isolated vertices 1 , . . . , n and no edges.An element g = ( g , . . . , g n ) ∈ G n with non-trivial intersection graph I ( g ) containing e ≥ α since contributionscoming from the 2 e − subgraphs of I ( g ) containing an even number of edgescancel out with contributions associated to the 2 e − subgraphs having anodd number of edges. ✷ Remark 3.4. Introducing α Γ = { g ∈ G n | I ( g ) = Γ } , we have α = α T where T denotes the trivial graph with n isolated vertices , . . . , n . Our proof of Proposition 3.3 computes α by applying M¨obius in-version (more precisely, its dual form, see Proposition 3.7.2 of [5]) α = X Γ ∈B µ (Γ) ♯ ( R Γ ) (7) (with µ (Γ) = ( − e (Γ) denoting the M¨obius function of the Boolean lattice B of all simple graphs on , . . . , n ) to the numbers ♯ ( R Γ ) = X Γ ′ ⊃ Γ α Γ ′ given by Proposition 3.1. Proof of Proposition 2.2: Combinatorics of genericpackings A hypergraph consists of a set V of vertices and of a set of hyperedges wherea hyperedge is a subset of V containing at least 2 vertices. Two verticesare adjacent if they belong to a common hyperedge. A path is a sequenceof consecutively adjacent vertices. A hypergraph is connected if any pair ofvertices can be joined by a path. A cycle is a closed path involving onlydistinct vertices. A hyperforest is a hypergraph with distinct hyperedgesintersecting in at most a common vertex and with every cycle contained ina hyperedge. A hypertree is a connected hyperforest.The primal graph of a hypergraph with vertices V is the ordinary graphwith vertices V and ordinary edges encoding adjacency in the hypergraph.An ordinary graph Γ is the primal graph of a hyperforest if and only ifevery cycle and every edge of Γ is contained in a unique maximal completesubgraph. Maximal complete subgraphs of such a graph Γ are in one-to-one correspondence with hyperedges of the associated hyperforest. Primalgraphs of hyperforests are often called block-graphs or chordal and diamond-free graphs . In the sequel, we identify generally hyperforests with theirprimal graphs. Lemma 4.1. The intersection g i S i ∩ g j S j associated to an edge { i, j } inan intersection graph I ( g ) is reduced to a unique element if S , . . . , S n is ageneric family of G . Proof Otherwise there exist two distinct elements a i , b i ∈ S j and twodistinct elements a j , b j ∈ S j such that g i a i = g j b j and g j a j = g i b i . Thisshows g i a i b − j g − j g j a j b − i g − i = e and implies the relation b − i a i b − j a j = e with b − i a i ∈ S − i S i \{ e } and b − j a j ∈S − j S j \ { e } in contradiction with genericity of the family S , . . . , S n . ✷ Proposition 4.2. Intersection graphs of generic families are (primal graphsof ) hyperforests. Proof Consider k cyclically consecutive vertices i , i , . . . , i k − , i k , i k +1 = i in an intersection graph I ( g ) of a generic family S , . . . , S n ⊂ G . Lemma4.1 implies the existence of unique elements a i j ∈ S i j and b i j +1 ∈ S i j +1 such that g i j a i j = g i j +1 b i j +1 for every edge { i j , i j +1 } of C . We get thus therelation g i a i ( g i b i ) − g i a i ( g i b i ) − · · · g i k a i k ( g i b i ) − = e which is conjugate to the relation (cid:0) b − i a i (cid:1) (cid:0) b − i a i (cid:1) · · · (cid:16) b − i k a i k (cid:17) = e . S , . . . , S n implies a i j = b i j for all j . The sets g i j S i j intersect thus in the common element g i a i = · · · = g i k a i k (which isthe unique common element of pairwise distinct sets in { g i S i , . . . , g i k S i k } by Lemma 4.1). All elements i , . . . , i k of I ( g ) are thus adjacent verticescontained in a common maximal complete subgraph of I ( g ).Suppose now that an edge { i, j } belongs to two distinct maximal com-plete subgraphs K and K ′ of I ( g ). Maximality of K and K ′ implies theexistence of vertices k ∈ K \ K ′ and k ′ ∈ K ′ \ K . Thus we get tripletsof mutually adjacent vertices i, j, k ⊂ K and i, j, k ′ ⊂ K ′ . Lemma 4.1shows that g i S i ∩ g j S j is reduced to a unique element a . We have thus g i S i ∩ g j S j ∩ g k S k = { a } ⊂ K . Similarly, we get a ∈ g k ′ S k ′ . This implies k ′ ∈ K in contradiction with k ′ ∈ K ′ \ K .Distinct maximal complete subgraphs of I ( g ) intersect thus at most ina common vertex and every cycle of I ( g ) is contained in a unique maximalcomplete subgraph of I ( g ). This implies that I ( g ) is (the primal graph of)a hyperforest. ✷ For the sake of concision, we identify in the sequel such an intersectiongraph I ( g ) with the corresponding hyperforest.Applying the proof of Proposition 4.2 to a Hamiltonian cycle visiting allvertices of a hyperedge { i , . . . , i k } in an intersection graph I ( g ) associatedto a generic family we get the following result: Proposition 4.3. Given a hyperedge { i , . . . , i k } in the intersection graph I ( g ) of a generic family S , . . . , S n , there exists a unique element a ∈ G such that g i l S i l ∩ g i m S i m = { a } for every pair of distinct vertices i l , i m in { i , . . . , i k } . Proposition 4.4. Let Γ be a hyperforest with vertices , . . . , n indexing thesubsets S , . . . , S n of a generic family in a group G . Defining the equivalencerelation E F of a hyperforest F as in Section 3.2, we have ♯ ( E F ) = n Y j =1 ( ♯ ( S j )) deg F ( j ) with deg F ( j ) denoting the degree of j defined as the number of distinct hy-peredges containing the vertex j . Proof Let e = { i , . . . , i k } be a hyperedge of an intersection graph I ( g ).Since T kj =1 g i j S i j is reduced to a unique element a e ∈ G , we get a map µ e : { i , . . . , i k } −→ G such that µ e ( i j ) ∈ S i j by setting µ e ( i j ) = g − i j a e .This map depends only on the equivalence class in E I ( g ) of I ( g ) and the setof all such maps determines the equivalence class of I ( g ) in E F for any hy-perforest F contained in I ( g ). Since all cycles of a hyperforest are containedin hyperedges, all possible choices of the maps µ e associated to hyperedges11f F correspond to equivalence classes of E F . Different choices yield in-equivalent classes. The set E F of all equivalence classes is thus in one-to-onecorrespondence with the set Q nj =1 S deg F ( j ) j . ✷ Proof of Proposition 2.2 Setting s i = ♯ ( S i ), Proposition 4.4 can berewritten as the identity ♯ ( E F ) = Y j =1 s deg F ( j ) j for every hyperforest F with vertices { , . . . , n } . We denote by HF ( n ) theset of all hyperforests with vertices { , . . . , n } . The set HF ( n ) is partiallyordered by inclusion by setting F ′ ≤ F for F ′ , F ∈ HF ( n ) if every hyperedgeof F ′ is contained in some hyperedge of F . Equivalently, F ′ ≤ F if adjacentvertices of F ′ are also always adjacent in F . The primal graph underlying F ′ is thus a subgraph of the primal graph underlying F if F ′ ≤ F . Denoting by µ the M¨obius function of the poset HF ( n ), the number α = α ( G ; S , . . . , S n )of packings of a generic family S , . . . , S n in a group G of order N is givenby α = X F ∈HF ( n ) µ ( F ) N c ( F ) n Y j =1 s deg F ( j ) j (8)(with c ( F ) denoting the number of connected components of a hyperforest F ). Since the M¨obius function of HF ( n + 1) restricts to the M¨obius functionof HF ( n ), the summation over HF ( n ) in Formula (8) can be extended (aftersetting s i = 0 for i > n and using the convention 0 = 1) over the poset HF of all hyperforests with vertices N \ { } such that almost all vertices areisolated (only finitely many vertices have strictly positive degree).Summing over all possible labellings of an unlabelled hyperforest andremarking that the M¨obius function is invariant under permutations of labelsshows that α is a symmetric function of s , . . . , s n . This expresses α as apolynomial in σ = P s i , σ = P i 1) = X k ( − k + j (cid:18) kj (cid:19) ( t i,k ( n ) + t i +1 ,k ( n )) (9)13here P k f ( k ) = P k ∈ Z f ( k ) since (cid:0) kj (cid:1) ( t i,k ( n ) + t i +1 ,k ( n )) = 0 for k < j or k > n − i . We prove (9) by induction on n . A straightforward computationshows that it holds n = 2. Applying the recursion relation (2) which holdsfor all i, j ∈ Z if n ≥ R = X k ( − k + j (cid:18) kj (cid:19) ( t i,k ( n ) + t i +1 ,k ( n ))of (9) we get R = X k ( − k + j (cid:18) kj (cid:19)(cid:16) ( i − t i − ,k ( n − 1) + t i − ,k − ( n − 1) + ( i − t i − ,k ( n − i − t i,k ( n − 1) + t i,k − ( n − 1) + ( i − t i − ,k ( n − (cid:17) = L + C where L = ( i − X k ( − k + j (cid:18) kj (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − X k ( − k + j − (cid:18) kj − (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − i − X k ( − k + j (cid:18) kj (cid:19) ( t i − ,k ( n − 1) + t i − ,k ( n − C = − X k ( − k + j − (cid:18) kj − (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − X k ( − k + j (cid:18) kj (cid:19) ( t i − ,k − ( n − 1) + t i,k − ( n − X k ( − k + j (cid:18) kj (cid:19) ( t i,k ( n − 1) + t i − ,k ( n − X k ( − k + j (cid:18) kj − (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − − X k ( − k + j (cid:18) k + 1 j (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − X k ( − k + j (cid:18) kj (cid:19) ( t i − ,k ( n − 1) + t i,k ( n − X k ( − k + j (cid:18)(cid:18) kj − (cid:19) − (cid:18) k + 1 j (cid:19) + (cid:18) kj (cid:19)(cid:19) ( t i − ,k ( n − 1) + t i,k ( n − C = 0 since (cid:0) k +1 j (cid:1) = (cid:0) kj − (cid:1) + (cid:0) kj (cid:1) .Using induction on n and applying (9) we get L = ( i − t i − ,j ( n − 1) + t i − ,j − ( n − t i − ,j − ( n − 1) + t i − ,j − ( n − i − t i − ,j ( n − 1) + t i − ,j − ( n − L = ( i − t i − ,j ( n − 1) + t i − ,j − ( n − 1) + ( i − t i − ,j ( n − i − t i − ,j − ( n − 2) + t i − ,j − ( n − 2) + ( i − t i − ,j − ( n − L = t i,j ( n ) + t i,j − ( n − ✷ Assuming the existence of two distinct series U , U fulfilling the require-ments of Proposition 2.5, the difference D = U − U = P ∞ n =1 D n x n satisfiesall hypotheses except for the value of its constant term. Since U and U are different, there exists a minimal natural integer n ≥ D n = 0.Let m ≥ n + 1 be the smallest integer such that D n = P nk = m σ k C n,k with C n,k ∈ C [ σ , σ , . . . ] and C n,m = 0. Since D n is of degree ≤ n with respectto the grading given by deg( σ i ) = i , we have C n,m ∈ C [ σ , . . . , σ n − m ] ⊂ C [ σ , . . . , σ n − ].Equation (6) and minimality of n imply D n (1 + σ , σ + σ , σ + σ , . . . ) = D n ( σ , σ , σ , . . . )or equivalently n X k = m ( σ k − + σ k ) C n,k (1+ σ , σ + σ , σ + σ , . . . ) = n X k = m σ k C n,k ( σ , σ , σ , . . . ) . Comparison of both sides modulo the ideal I generated by σ m , σ m +1 , σ m +2 , . . . gives C n,m (1 + σ , σ + σ , σ + σ , . . . ) = 0 . Algebraic independency of the symmetric functions σ , σ , . . . shows thus C n,m = 0 in contradiction with our assumption. ✷ .15 The M¨obius function for the poset of finite la-belled hyperforests Let P be a poset (partially ordered set) such that P has a unique minimalelement m and { y ∈ P | y < x } is finite for all x ∈ P . This allows therecursive definition of a M¨obius function µ by setting µ ( m ) = 1 and µ ( x ) = − P y The M¨obius function µ ( F ) of a hyperforest F in the poset HF of all vertex-labelled hyperforests with finitely many hyperedges is givenby µ ( F ) = Y j ≥ ( − ( j − κ j (10) where κ j denotes the number of hyperedges involving exactly j vertices of F . Remark 7.2. The poset HF is in fact a lattice with wedge F ∧ F given bythe intersection and join F ∨ F given by the smallest hyperforest containing F and F . Proof of Proposition 7.1 Remark first that the order relation inducedon subforests of a given hyperforest F ∈ HF is the product order of allorder-relations on hyperedges of F . An easy argument (or Proposition 3.8.2of [5]) shows thus that we have µ ( F ) = Y e ∈ E ( F ) µ ( e )16here E ( F ) denotes the set of hyperedges of F and where µ ( e ) is the M¨obiusfunction restricted to a hyperedge e ∈ E ( F ). This can of course be rewrittenas µ ( F ) = Y j ≥ µ ( K j ) κ j where K j is an abitrary hyperedge on j labelled vertices and where κ j is thenumber of hyperedges having j vertices of F .The proof of Proposition 2.2 shows that µ ( K j ) coincides with the coeffi-cient of σ j +1 x j in U . By Theorem 2.1 (whose proof needs only the existencebut not the exact determination of the M¨obius function), this coefficientequals − t j +1 , ( j ) = − ( j − t i,j ( n ) recursively. ✷ Remark 7.3. It would be interesting to have a simple direct proof that µ ( K n ) = − ( n − for a hypergraph K n ∈ HF reduced to a unique hyperedgeinvolving n ≥ vertices. Remark 7.4. Let HT k ( n ) be the finite set of all hypertrees with k hyperedgesand n labelled vertices. Denoting by ♯ ( e ) the number of vertices involved ina hyperedge e , we have X T ∈HT k ( n ) Y e ∈E ( T ) ( ♯ ( e ) − n Y j =1 s deg( j ) j = ( − n + k +1 σ n σ k − S ( n − , k ) (11) where σ = P nj =1 s j and σ n = Q nj =1 s j and where S ( n, k ) denotes theStirling number of the first kind defined by n X k =0 S ( n, k ) x k = x ( x − x − · · · ( x − n + 1) = n − Y j =0 ( x − j ) . Indeed, the proof of Proposition 2.2 shows that a hyperforest with n non-isolated vertices, k hyperedges and c connected components yields only con-tributions to the coefficients of x n − c σ n + s σ k − − s for s = 0 , . . . , k − . The co-efficient of x n − σ n σ k − in U is thus obtained from contributions from all ele-ments in the set HT k ( n ) of hypertrees with n non-isolated vertices { , . . . , n } and k hyperedges. This coefficient equals ( − n +1 σ n σ k − S ( n − , k ) by For-mulae (4) and (3). Formulae (8) and (10) show that a hypertree T ∈ HT k ( n ) contributes a summand given by ( − k Q e ∈E ( T ) ( ♯ ( e ) − Q nj =1 s deg( j ) j to thecoefficient of x n − σ n σ k − in U .Setting s = · · · = s n = 1 , Formula (11) specializes to the identity X T ∈HT k ( n ) Y e ∈E ( T ) ( ♯ ( e ) − n k − S ( n − , k )( − n + k +1 hich is analogous to a Theorem of Husimi (see [2] or [1]) expressing thetotal number n k − S ( n − , k ) of elements in the set HT k ( n ) of labelled hypertrees with k hyperedges and n vertices in terms of Stirling numbers of the second kind.All these results are of course generalizations and variations of Cayley’stheorem corresponding to the case k = n − and showing that there are n n − labelled trees on n vertices.Observe that all these identities can also be deduced for example from Ex-ercice 5.30 in [5] using a well-known map between hypergraphs and ordinarybipartite graphs. The computation of U ( x, σ , σ , . . . ) up to o ( x n ) is straightforward using therecurrence relation (2). For a given fixed numerical value of σ , the followingtrick reduces memory requirement and speeds the computation up: Setting c n ( σ ) = ( γ n +1 ( σ , n ) , γ n +2 ( σ , n ) , . . . , γ n ( σ , n ))with γ i ( σ , n ) = P n − ij =0 t i,j ( n )( − σ ) j we have U ( x, σ , σ , . . . ) = 1 − ∞ X n =1 h c n ( σ ) , ( σ n +1 , . . . , σ n ) i x n where h a, b i = P i ∈ I a i b i for two finite-dimensional vectors a, b with coeffi-cients indexed by a common finite set I . The coefficients γ i ( σ , n ) of c n ( σ )can be computed from the coefficients of c n − ( σ ) by the formula γ i ( σ , n ) = ( i − − σ ) γ i − ( σ , n − 1) + ( i − γ i − ( σ , n − 1) (12)with missing coefficients omitted in the case of i = n + 1 or i = 2 n .The coefficients of the first vectors c (0) , c (0) , c (0) , . . . are given by therows of 11 12 5 36 26 35 1524 154 340 315 105 , see A112486 of [4]. 18 .1 The examples U ( x, − , − , − , . . . ) and U ( x, , − , − , − , . . . ) The series U ( x, − , − , − , − , . . . ) − S ( n ) = X i,j t i,j ( n )enumerating the sums of the triangles T ( n ) defined by the integers t i,j ( n ).We have (1 + x ) U ( x, − , − , − , − , . . . )= U ( x, , − , − , − , − , . . . )= 2 U ( x, , − , − , − , − , . . . ) − U ( x, , − , − , − , . . . ) − s ( n ) = n X i = n +1 t i, ( n )starting as1 , , , , , , , , , , . . . , cf. A112487 of [4], and obtained by summing the integers of the first columnof the triangles T (1) , T (2) , . . . . In particular, we have 2 s ( n ) = S ( n − S ( n )or equivalently2 n X i = n +1 t i, ( n ) = n X i = n +1 2 n − i X j =0 t i,j ( n ) + n − X i = n n − − i X j =0 t i,j ( n − n ≥ The recursive definition of the integers t i,j ( n ) implies easily that specializa-tions of the form σ n = c Q Ak =1 ( n + a k )! Q Bl =1 ( n + b l )! z λn + r , n ≥ , or σ n = c Q Ak =1 ( n + a k )! Q Bl =1 ( n + b l )! e ( λn + r ) z , n ≥ , (with λ = 0 and b i 6∈ {− , − , − , − , . . . } ) lead to differential equationswith respect to z for rational expressions of U ( x, − y, σ , σ , σ , . . . ). Such aseries U is analytic if B > A . We illustrate this with the following examples.19 .2.1 U ( x, − y, − z r , − z r , − z r , . . . )Setting σ n = − z n + r for n = 2 , , . . . the series f ( z ) = U ( x, − y, σ , σ , . . . ) − f = xz (cid:18) z r +1 + ( y − (1 + z )(1 + r )) f + z (1 + z ) dfdz (cid:19) . U (cid:16) x, − y, − z r (2+ b )! , − z r (3+ b )! , − z r (4+ b )! , . . . (cid:17) Setting σ n = − z n + r ( n + b )! for n = 2 , , . . . the series f ( z ) = U ( x, − y, σ , σ , . . . ) − b − r )( b − − r ) f + 2( b − r ) zf ′ + z f ′′ = xz r b ! + xz ( r − b − z ( r + 1) + ( r − b )( r − y )) f + xz ( b + y + z − r ) f ′ + xz f ′′ U (cid:0) x, − y, − e (2 λ + r ) z , − e (3 λ + r ) z , − e (4 λ + r ) z , . . . (cid:1) Setting σ n = − e ( nλ + r ) z for n = 2 , , . . . , the series f ( z ) = U ( x, − y, σ , σ , σ , . . . ) − f = xe λz (cid:18) e ( λ + r ) z + (cid:16) y − (cid:16) rλ (cid:17) (cid:16) e λz (cid:17)(cid:17) f + 1 + e λz λ dfdz (cid:19) . U (cid:0) x, − y, − (2 + a )! e (2 λ + r ) z , − (3 + a )! e (3 λ + r ) z , − (4 + a !) e (4 λ + r ) z , . . . (cid:1) Setting σ n = − ( n + a )! e ( nλ + r ) z for n = 2 , , . . . , the series f ( z ) = U ( x, − y, σ , σ , . . . ) − f = xe λz λ ( λ (1 + a ) − r ) (cid:16) λ ( λ ( y − − r ) + ( λ + r )( r − λ (2 + a )) e λz (cid:17) f + xe λz λ (cid:16) λ ( λ ( y + a ) − r ) + (3 r − λr ( a + 1) + λ ( a + a − e λz (cid:17) f ′ + xe λz λ (cid:16) λ + (2 λ (1 + a ) − r ) e λz (cid:17) f ′′ + xe λz λ f ′′′ + (2 + a )! xe (2 λ + r ) z .2.5 U (cid:16) x, − y, − e (2 λ + r ) z (2+ b )! , − e (3 λ + r ) z (3+ b )! , − e (4 λ + r ) z (4+ b )! . . . (cid:17) Setting σ n = − e ( nλ + r ) z ( n + b )! for n = 2 , , . . . the series f ( z ) = U ( x, − y, σ , σ , . . . ) − λb − r )( λ ( b − − r ) f + ( λ (2 b − − r ) f ′ + f ′′ = x λ b ! e (2 λ + r ) z + xe λz (cid:16) ( λb − r )( λ ( y − − r ) − λ ( λ + r ) e λz (cid:17) f + xe λz (cid:16) λ ( b − y ) − r + λe λz (cid:17) f ′ + xe λz f ′′ Remark 8.1. The recursion relation (2) gives rise to partial differentialequations for generating series of t i,j ( n ) which are exponential with respectto j and/or n . Proposition 8.2. Let σ , σ , . . . be a sequence of complex numbers of theform σ n = ( − n P ( n ) for all n ≥ A where A is some natural integer andwhere P ( s ) ∈ C [ s ] is a polynomial. Then U ( x, σ , σ , . . . ) is a rational series. Proof Let d denote the degree of P . Applying identity (6) of Proposi-tion 2.3 iteratively d +1 times we get a series of the form U ( x, ˜ σ , ˜ σ , . . . , ˜ σ A + d +2 , , , , . . . )which is a polynomial. ✷ As an illustration we consider the series U ( x, y, , − , , . . . ). Proposi-tion 2.3 shows(1 − xy ) U ( x, y, , − , , − , . . . ) = U ( x, y, y, , , . . . ) = 1 − (1 + y ) x . We have thus U ( x, y, , − , , . . . ) = 1 − x − xy . U ( x, σ , P (2) , P (3) , P (4) , . . . ) Proposition 8.3. Let P ( s ) ∈ C [ s ] be a polynomial of degree d . There existconstants α , . . . , α d ∈ C such that [ x n ] U ( x, σ , P (2) , P (3) , P (4) , . . . ) = d X h =0 α h [ x n + h ] U ( x, σ , , , , , , . . . ) for all n ≥ with [ x n ] U denoting the coefficient of x n in the series U . Proof The proof is by induction on d and holds certainly for d = 0.Setting γ i ( n ) = P nj = n +1 t i,j ( n )( − σ ) j , formula (12) implies0 = − i d γ i ( n + 1) + i d ( i − − σ ) γ i − ( n ) + i d ( i − γ i − ( n )= − i d γ i ( n + 1) + ( i − d +1 γ i − ( n ) + ( i − d +1 γ i − ( n ) ++ Q ( i − γ i − ( n ) + Q ( i − γ i − ( n )21here Q and Q are polynomials of degree ≤ d . Fixing n and summingover i we get2 n X i = n +1 i d +1 γ i ( n ) = n +2 X i = n +2 i d γ i ( n + 1) − n X i = n +1 ( Q + Q )( i ) γ i ( n ) . (13)The right side of (13) equals now[ x n +1 ] U ( x, σ , d , d , d , . . . ) − [ x n ] U ( x, σ , ( Q + Q )(2) , ( Q + Q )(3) , . . . ) . It is thus by induction on d a linear combination of the coefficients of x n , . . . , x n + d +1 in U ( x, σ , , , , . . . ). This proves the result for U ( x, σ , d +1 , d +1 , . . . ).The general induction step follows by remarking that all coefficients ofstrictly positive degree in x of U ( x, σ , σ , . . . ) are linear in σ , σ , . . . . ✷ s (1) , s (2) , . . . Computations with a few thousand values of s ( n ) suggest the followingasymptotic formula for the integral sequence s ( n ) = P ni = n +1 t i, ( n ): Conjecture 9.1. There exists a sequence A , A , . . . of rational polynomials A i ( x ) ∈ Q [ x ] with A i of degree i such that s ( n ) = n n − (1 − log 2) n − / e n m X k =0 A k (1 − log 2) n k + o ( n − m ) ! for all m ∈ N . The first few polynomials A , A , . . . are A = 1 A = 1124 − x A = 2651152 − x 288 + x A = 48703414720 − x x x A = 233371739813120 − x x x − x A = 381807611337720832 − x x x − x − x B k of the formal power series P ∞ k =1 B k ( x ) t k = log (cid:0)P ∞ k =0 A k ( x ) t k (cid:1) seem to be simpler and start as B = 1124 − x B = 18 − x B = 1272880 − x 16 + x 288 + x B = 164 − x 32 + 11 x 576 + x B = 22140320 − x 64 + 41 x 576 + 1381 x − x − x Remark 9.2. The constant − log 2 = . . . . appearing in Con-jecture 9.1 seems also to be related to the index m n such that t m n , ( n ) =max i ( t i, ( n )) with m n given asymptotically by n − log 2) . Moreover, we haveseemingly lim n →∞ t mn, ( n ) √ ns ( n ) ∼ . (and the numbers t i, ( n ) , suitably rescaled,should satisfy a central limit Theorem). 10 Modular properties of the sequence s (1) , s (2) , . . . Proposition 10.1. The series U ( x, σ , σ , . . . ) ∈ F p [[ x ]] is rational if σ , σ , . . . is an ultimately periodic sequence of elements in F p . Proof Up to addition of a polynomial to U = U ( x, σ , σ , . . . ) we can sup-pose that σ , σ , σ , . . . is periodic with period k . We set ˜ σ i = σ i for i ≥ σ , ˜ σ , . . . , to a k − periodic sequence indexed by Z . We supposefirst σ = 0 in F p . Using the identity ( − σ ) p − = 1 and periodicity of thesequence (˜ σ i ) i ∈ Z , we have U = 1 − ∞ X n =1 x n pk − X i =0 X α ∈ Z ˜ σ i + αkp p − X j =0 X β ∈ Z t i + αkp,j + β ( p − ( n )( − σ ) j + β ( p − = 1 − ∞ X n =1 x n pk − X i =0 ˜ σ i p − X j =0 ( − σ ) j X α ∈ Z X β ∈ Z t i + αkp,j + β ( p − ( n ) . Since the recurrence relations (2) define the elements t i,j (2) , t i,j (3) , . . . cor-23ectly for arbitrary indices i, j ∈ Z we have X α ∈ Z X β ∈ Z t i + αkp,j + β ( p − ( n )= ( i − X α ∈ Z X β ∈ Z t i − αkp,j + β ( p − ( n − X α ∈ Z X β ∈ Z t i − αkp,j − β ( p − ( n − i − X α ∈ Z X β ∈ Z t i − αkp,j + β ( p − ( n − n ≥ 2. Setting ˜ t i,j ( n ) ≡ X α ∈ Z X β ∈ Z t i + αkp,j + β ( p − ( n − ≤ i < kp and 0 ≤ j < p − 1, the elements ˜ t i,j ( n ) of F p satisfy therecursion relation (2) with indices considered modulo kp for i and modulo p − j . Since the kp ( p − 1) elements ˜ t i,j ( n ) of the finite field F p dependaffinely on the kp ( p − 1) elements ˜ t i,j ( n − 1) for n ≥ 2, finiteness of the set( i, j ) of indices implies the existence of an integer l such that ˜ t i,j ( n + l ) = t i,j ( n ) for all sufficiently large n and for all possible indices i and j . Thisimplies easily that the coefficients of U are ultimately periodic and ends theproof for σ = 0.The case σ = 0 involves only the integers t i, ( n ) and their analogues˜ t i, ( n ) with indices in the finite set { , . . . , pk − } . Details are similar tothe previous case and left to the reader. ✷ The first non-trivial case of Proposition 10.1 is perhaps given by the gen-erating series U ( x, , − , − , − , . . . ) with coefficients of U ( x, , − , − , . . . ) − s ( n ) = n X i = n +1 t i, ( n )obtained by summing all coefficients in the first column of the triangulararrays T (1) , T (2) , . . . . Conjecture 10.2. There exists a sequence α = − , α = 2 , α = 0 , α = 13 , α = 518 , α = 149540 , α = 5532025 ,α = 18497416804000 , α = 7751671192857680000 , α = 3252149573711200225600000 , . . . of rational numbers such that (cid:0) x p − (cid:1) ∞ X n =1 s ( n ) x n ≡ x + p − X n =0 α n x p − n (mod p )24 or every prime number p . Conjecture 10.3. The rational sequence α , α , . . . has an asymptotic ex-pansion given by α n ∼ ∞ X k =1 k k − n k ! (cid:18) e (cid:19) k and converges with limit given by e − = . . . . .The error term ǫ n = α n − ∞ X k =1 k k − n k ! (cid:18) e (cid:19) k is given by ǫ n = ( − n +1 (1 − log 2) s ( n + 1) m X k =0 γ k (1 − log 2) n k + o (cid:0) n − m − (cid:1)! where γ k ( x ) ∈ Q [ x ] is a polynomial of degree at most k . The first fewpolynomials are given by γ = 1 γ = − x γ = − x 48 + x 48 + x γ = − x 192 + 5 x 96 + 193 x − x − x 11 Integer sequences obtained as weighted sumsof the numbers t i,j ( n ) The sequence q i ( x, y ) defined by q i ( x, y ) = X n,j ≥ t i,j ( n ) x n y j = i − X n = ⌈ i/ ⌉ x n n − X j =0 t i,j ( n ) y j is given by q ( x, y ) = 0 , q ( x, y ) = x and by the recursion relation q i ( x, y ) = x (( i − y ) q i − ( x, y ) + ( i − q i − ( x, y ))for i ≥ 3. The following table lists the first few non-zero coefficients q ( x, y ) , q ( x, y ) , . . . (up to normalizations) and the seemingly corresponding25equences of [4] (which have often interesting combinatorial interpretations)for a few specializations: q i ( − , − / − i +1 / , , , , , , . . . A q i ( − , / − i +1 / , , , , , , . . . A q i ( − , − / − i +1 / , − , − , − , − , − , − , . . .q i ( − , − / − i +1 / , , , , , , . . . A q i ( − , / − i +1 / , , , , , , . . . A q i ( − , − − i +1 , , − , − , − , − , − , . . .q i ( − , − i +1 , , , , , , . . . A q i ( − , − i +1 , , , , , , , . . . A q i ( − , − i +1 , , , , , , . . . A q i ( − , − i +1 , , , , , , . . . A q i (1 , − − i , , , , , , . . . A q i (1 , − 1) 1 , , , , , , , , . . . A q i (1 , 0) 1 , , , , , , , . . . A q i (1 , 1) 1 , , , , , , . . . A q i (1 , 2) 1 , , , , , , . . . A q i (1 , 3) 1 , , , , , , . . . A q i (2 , − / − i / , , , , , , , , . . . A q i (2 , − / − i / , , , , , , . . . A q i (2 , − / − i / , , , , , , . . . A q i (2 , − / , , , , , , , . . . A q i (2 , − / / , , , , , , . . . A q i (3 , − / − i / , , , , , , . . . A q i (3 , − / , , , , , , . . . A q i (1 , , q i (1 , , q i (1 , q i (1 , 3) can seemingly also be obtained by considering the weighted sums s i = X j,n ≥ (cid:18) n − j + k − k (cid:19) t i,j ( n )( − j for k = 1 , , q i (cid:18) κ, − κ + 1 κ (cid:19) ( − i κ and 1 + ( i − κ = q i (cid:18) κ, − κ + 1 κ (cid:19) ( − i κ which hold for κ = 0 and for i = 2 , , , . . . and which can easily be proven byinduction. (These two examples generalize probably to q i (cid:0) κ, − λκ +1 κ (cid:1) ( − i κ =26 λ ( i, κ ) , λ = 1 , , , . . . , κ = 0 , i = 2 , , , . . . for P λ a suitable polynomialfunction of κ and i .)Another identity is given by the family of weighted examples1 = ( − i k !( i − k ) X j,n ≥ t i,j ( n ) ( − j ( n + k − j )!for all k ∈ N and for all i ≥ a i = X j,n ≥ ( n − j ) t i,j ( n )( − j ,b i = 14 X j,n ≥ ( n − − j ) t i,j ( n )2 n (cid:18) − (cid:19) j ,c i = X j,n ≥ t i,j ( n )( n − − j )!( − j ,d i = ( − i +1 X j,n ≥ t i,j ( n )( n − − j )!( − n ,e i = X j,n ≥ t i,j ( n ) ( − j ( n − − j )! ,f i = ( i − X n ≥ t i, ( n )( n − . Their initial coefficients (with leading zeros omitted) and the seemingly cor-responding sequences of [4] are as follows: a i , , , , , , , , , . . . A b i , , , , , , . . . A c i , , , , , , . . . A d i , , , , , , . . . A e i , , , , , , , . . . A f i , , , , , , . . . A 12 Coverings Coverings and packings are dual notions. We discuss here a few aspects ofthe theory of coverings in relation with packings by generic families.27 (left-)covering with parts S , . . . , S n of a group G is a vector ( g , . . . , g n )such that G = ∪ nj =1 g j S j .A covering of a finite group G with N elements by non-empty subsets S , . . . , S n exists of course always if n ≥ N .We are interested in large collections of subsets S , . . . , S n in a finitegroup G of order N such that the sets S , . . . , S n (or more precisely, suitabletranslates) cover G and the number of all coverings depends only on thecardinalities of S , . . . , S n (and of N ) for every family S , . . . , S n in thecollection.Three such collections can be described as follows:Start with a family S , . . . , S n which is generic for packings and add N − P nj =1 ♯ ( S j ) singletons. Coverings of G by such families are “tight”and essen-tially in one-to-one correspondence (except for a factor (cid:16) N − P nj =1 ♯ ( S j ) (cid:17) !accounting for all permutations of the added singletons) with packings by S , . . . , S n .The second family is obtained by adding N + n − − P nj =1 ♯ ( S j ) singletonsto a family S , . . . , S n which is generic for packings. The fact that thenumber of associated coverings depends only on all involved cardinalities issimilar to the proof of Proposition 2.2 given in Section 4. The proof needsprobably computations with the full M¨obius function. I do not know ifthere is an efficient way for computing the number of associated coveringsor if there is a nice formula similar to the one assciated to enumerations ofpackings.There is a further variation on this theme: Given an arbitrary natu-ral integer a one can consider adding N + a − P nj =1 ♯ ( S j ) singletons to afamily S , . . . , S n which is generic for packings. For every natural integer a , such a family has the property that the number of associated coveringsdepends only on all involved cardinalities. The choices a = 0 correspondingto the first family and a = n − g S , . . . , g n S n is a hyperforest, the union ∪ nj =1 g j S j contains at least P nj =1 ♯ ( S j ) − ( n − 1) elements. This leaves at most N + n − − P nj =1 ♯ ( S j )missing elements which can be covered using the additional singletons.A third rather trivial family is given by considering complements G \S , . . . , G \ S n where S , . . . , S n is a generic family for packings in G havingat least two parts. The number of coverings of such a family is easy tocompute and given by N n − N Q nj =1 ♯ ( S j ).It would perhaps be interesting to have other (and hopefully more exotic)families of examples. Acknowledgements. I thank Pierre de la Harpe for helpful comments andtwo anonymous referees for their careful work and useful remarks.28 eferences [1] I.M. Gessel, L.H. Kalikow, Hypergraphs and a functional equation ofBouwkamp and de Bruijn. J. Combin. Theory Ser. A 110 (2005), no. 2,275–289.[2] K. Husimi, Note on Mayer’s theory of cluster integrals , Journal ofChemical Physics (1950), 682–684.[3] J. McCammond, J. Meier, The hypertree poset and the l -Betti numbersof the motion group of the trivial link. Math. Ann.328