Gončarov Polynomials in Partition Lattices and Exponential Families
aa r X i v : . [ m a t h . C O ] J u l Gonˇcarov Polynomials in Partition Latticesand Exponential Families
Dedicated to Joseph KungAyomikun Adeniran ∗ and Catherine Yan † Department of Mathematics, Texas A&M University, College Station, TX 77843
Abstract
Classical Gonˇcarov polynomials arose in numerical analysis as a basis for the solutions ofthe Gonˇcarov interpolation problem. These polynomials provide a natural algebraic tool in theenumerative theory of parking functions. By replacing the differentiation operator with a deltaoperator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introducedthe sequence of generalized Gonˇcarov polynomials associated to a pair ( ∆ , Z) of a delta operator∆ and an interpolation grid Z . Generalized Gonˇcarov polynomials share many nice algebraicproperties and have a connection with the theories of binomial enumeration and order statistics.In this paper we give a complete combinatorial interpretation for any sequence of generalizedGonˇcarov polynomials. First we show that they can be realized as weight enumerators in par-tition lattices. Then we give a more concrete realization in exponential families and show thatthese polynomials enumerate various enriched structures of vector parking functions. Keywords : Gonˇcarov polynomials, partition lattices, exponential family
AMS Classification : 05A10, 05A15, 05A18, 06A07
The classical Gonˇcarov interpolation problem in numerical analysis was introduced by Gonˇcarov[2, 3] and Whittaker[17]. It asks for a polynomial f ( x ) of degree n such that the i th derivative of f ( x ) at a given point a i has value b i for i = , , , ..., n . The solution is obtained by taking linearcombinations of the (classical) Gonˇcarov polynomials, or the Abel-Gonˇcarov polynomials, whichhave been studied extensively by analysts; see e.g. [2, 10, 1, 4]. Gonˇcarov polynomials also playa crucial role in Combinatorics due to their close relations to parking functions. A (classical) ∗ Corresponding author: [email protected] † [email protected] arking function is a sequence ( a , a , ..., a n ) of positive integers such that for every i = , , ..., n ,there are at least i terms that are less than or equal to i . For example, the sequences ( , , , ) and ( , , , ) are both parking functions while ( , , , ) is not. The set of parking functionsstays in the center of enumerative combinatorics, with many generalizations and connectionsto other research areas, such as hashing and linear probing in computer science, graph theory,interpolation theory, diagonal harmonics, representation theory, and cellular automaton. Seethe comprehensive survey [19] for more on the combinatorial theory of parking functions.The connection between Gonˇcarov polynomials and combinatorics was first found by JosephKung, who in a short note [8] of 1981 proved that classical Gonˇcarov polynomials give theprobability distribution of the order statistics of n independent uniform random variables, andits difference analog describes the order statistics of discrete, injective functions. These resultswere further developed in [9] to an explicit correspondence between classical Gonˇcarov polyno-mials and vector parking functions. Inspired by the rich theory on delta operators and finiteoperator calculus, which is a unified theory on linear operators analogous to the differentiationoperator D and special polynomials, Lorentz, Tringali, and the second author of the presentpaper introduced the generalized Gonˇcarov polynomials [11] as a basis for the solutions to theGonˇcarov interpolation problem with respect to a delta operator. Many algebraic and analyticproperties of classical Gonˇcarov polynomials have been extended to the generalized version.A natural question is to find the combinatorial interpretations for the generalized Gonˇcarovpolynomials. To answer this question we need to understand the combinatorial significance ofdelta operators. In the third paper of the seminal series On the Foundations of CombinatorialTheory III , Mullin and Rota [12] developed the basic theory of delta operators and their asso-ciated sequence of polynomials. Such sequences of polynomials are of binomial type and occurin many combinatorial problems when objects can be pieced together out of small, connectedobjects. Mullin and Rota’s work provides a realization of binomial sequences in combinatorialproblems. However, this realization is only valid for binomial sequences whose coefficients arenon-negative integers, and so excludes many basic counting polynomials, for example, the fallingfactorial x ( n ) = x ( x − )⋯( x − n + ) . Mullin and Rota hint at a generalization of their theoryto incorporate such cases. Using the language of partitions, partition types and partition cat-egories, Ray [14] proved that every polynomial sequence of binomial type can be realized as aweighted enumerator in partition lattices.In this paper we give a complete combinatorial interpretation of the generalized Gonˇcarovpolynomials, first in Ray’s partition lattices and then in a more concrete model, the exponentialfamilies, as described by Wilf [18]. Basically, to any polynomial sequence of binomial type andany given interpolation grid Z there is an associated sequence of Gonˇcarov polynomials. Whilethe sequence of binomial type can be realized as weighted enumerators in partition lattices or inan exponential family, the associated Gonˇcarov polynomials count those structures which alsoencode vector parking functions. In other words, the generalized Gonˇcarov polynomials charac-terize structures in a binomial enumeration problem that are subject to certain order-statisticconstraints. Our results cover the initial attempt in [11] which provides a combinatorial inter-pretation for some families of generalized Gonˇcarov polynomials in a structure called reluctantfunctions .The rest of the paper is organized as follows. In Section 2, we recall the basic theory f delta operators and binomial enumeration, as well as the concepts of generalized Gonˇcarovpolynomials and vector parking functions. In Section 3, we describe the realization of generalizedGonˇcarov polynomials in partition lattices and weight functions. Then, in Section 4, we study amore concrete realization of Gonˇcarov polynomials as type-enumerator in exponential families.We end the paper in Section 5 with a few closing remarks. We recall the basic theory of delta operators and their associated sequence of basic polynomialsas developed by Rota, Kahaner, and Odlyzko [15]. Let K be a field of characteristic zero and K [ x ] the vector space of all polynomials in the variable x over K . For each a ∈ K , let E a denote the shift operator K [ x ] → K [ x ] ∶ f ( x ) ↦ f ( x + a ) . A linear operator s ∶ K [ x ] → K [ x ] iscalled shift-invariant if s E a = E a s for all a ∈ K , where the multiplication is the composition ofoperators. Definition 1.
A delta operator ∆ is a shift-invariant operator satisfying ∆ ( x ) = a for somenonzero constant a . Definition 2.
Let ∆ be a delta operator. A polynomial sequence { p n ( x )} n ≥ is called thesequence of basic polynomials, or the associated basic sequence of ∆ if(i) p ( x ) = p n ( x ) is n and p n ( ) = n ≥ ( p n ( x )) = np n − ( x ) .Every delta operator has a unique sequence of basic polynomials, which is a sequence ofbinomial type (or binomial sequence) that satisfies p n ( u + v ) = ∑ i ≥ ( ni ) p i ( u ) p n − i ( v ) , (1)for all n ≥
0. Conversely, every polynomial sequence of binomial type is the associated basicsequence of some delta operator.Let s be a shift-invariant operator, and ∆ a delta operator. Then s can be expanded uniquelyas a formal power series of ∆. If s = ∑ k ≥ a k k ! ∆ k , we say that f ( t ) = ∑ k ≥ a k k ! t k is the ∆-indicator of s . In fact, the correspondence f ( t ) = ∑ k ≥ a k k ! t k ←→ ∑ k ≥ a k k ! ∆ k is an isomorphism from the ring K [[ t ]] of formal power series in t onto the ring of shift-invariantoperators. Under this isomorphism, a shift-invariant operator is invertible if and only if its ∆-indicator f ( t ) satisfies f ( ) ≠
0, and it is a delta operator if and only if f ( ) = f ′ ( ) ≠ f ( t ) has a compositional inverse g ( t ) satisfying f ( g ( t )) = g ( f ( t )) = t . nother important result is the generating function for the sequence of basic polynomials { p n ( x )} n ≥ associated to a delta operator ∆. Let f ( t ) be the D -indicator of ∆, where D = d / dx is the differentiation operator. Let g ( t ) be the compositional inverse of f ( t ) . Then, ∑ n ≥ p n ( x ) t n n ! = exp ( xg ( t )) . (2)The operator Λ = g ( D ) is called the conjugate delta operator of ∆, and { p n ( x )} n ≥ is the conjugate sequence of Λ. It is easy to see that if p n ( x ) = ∑ k ≥ p n,k x k , then g ( t ) = ∑ k ≥ p k, t k k ! .Polynomial sequences of binomial type are closely related to the theory of binomial enumer-ation. Consider the following model. Assume B is a family of discrete structures. For a finiteset E , let Π ( E ) be the poset of all partitions π of E , ordered by refinement, and write ∣ π ∣ for thenumber of blocks of π . Define a k -assembly of B -structures on E as a partition π of the set E into ∣ π ∣ = k blocks such that each block of π is endowed with a B -structure. Let B k ( E ) denotethe set of all such k -assemblies. For example, when B is a set of rooted trees, a k -assembly of B -structures on E is a forest of k rooted trees with vertex set E . We can also take B to be otherstructures, such as permutations, complete graphs, posets, etc. Assume that the cardinality of B k ( E ) depends only on the cardinality of E , but not its content. In other words, there is abijection between B k ( E ) and B k ([ n ]) where [ n ] = { , , ..., n } and ∣ E ∣ = n . Definition 3.
Let b n,k = ⎧⎪⎪⎨⎪⎪⎩ ∣ B k ([ n ])∣ , if k ≤ n , if k > n, where b , = b n, = n ≥ Theorem 1 ([12]) . Assume b , ≠ . If b n ( x ) = ∑ nk = b n,k x k is the enumerator for assemblies of B -structures on [ n ] , then ( b n ( x )) n ≥ is a sequence of polynomials of binomial type. Theorem 1 provides a realization of binomial sequences in combinatorial problems. If wethink of x as a positive integer such that ∣ X ∣ = x for some set X , then we can interpret b n ( x ) asthe number of assemblies of B -structures on [ n ] , where each block carries a label from X . Fromthis viewpoint, it is easy to see that ( b n ( x )) n ≥ is of binomial type.This realization is only valid for binomial sequences whose coefficients are non-negativeintegers, and so excludes many polynomial sequences naturally appearing in combinatorics, forexample, the falling factorials x ( n ) . Mullin and Rota expanded their construction slightly byconsidering the monomorphic classes , in which different blocks receive different labels from X ,and hence the counting polynomial becomes ˜ b n ( x ) = ∑ nk = b n,k x ( k ) . Ray [14] extended Mullin-Rota’s theory and developed the concept of partition categories, and he proved that any binomialsequence can be realized as a weight enumerator in partition lattices. We will use Ray’s modelin Section 3. Let
Z = ( z i ) i ≥ be a fixed sequence with values in K , where K is the scalar field. For ourpurpose, it suffices to take K to be Q , R , or C . We call Z the interpolation grid and z i ∈ Z the -th interpolation node. Let T = ( t n ( x ; ∆ , Z )) n ≥ be the unique sequence of polynomials thatsatisfies ε z i ∆ i ( t n ( x ; ∆ , Z )) = n ! δ i,n , (3)where ε z i is evaluation at z i . Definition 4.
The polynomial sequence
T = ( t n ( x ; ∆ , Z )) n ≥ determined by (3) is called the sequence of generalized Gonˇcarov polynomials associated with the pair ( ∆ , Z ) and t n ( x ; ∆ , Z ) is the n -th generalized Gonˇcarov polynomial relative to the same pair.This sequence T has a number of interesting algebraic properties. One of them is a recurrenceformula described as follows: Let t n ( x ) = t n ( x ; ∆ , Z ) and { p n ( x )} n ≥ be the basic sequenceassociated to ∆. Then p n ( x ) = n ∑ i = ( ni ) p n − i ( z i ) t i ( x ) . (4)We remark that by definition, to compute the generalized Gonˇcarov polynomials given thebasic sequence, one would find the conjugate operator Λ via (2), compute ∆ by solving for thecompositional inverse of the D -indicator of Λ, and then find the n -th polynomial t n ( x ) of thesequence by using (3). The computation required in this process can be quite involved. However,(4) gives a recursive formula which can be used as an alternative definition for t n ( x ) , which ismuch more convenient in combinatorial problems. For other algebraic properties of generalizedGonˇcarov polynomials, see [11].Classical Gonˇcarov polynomials have a combinatorial interpretation in enumeration. Let ⃗ u = ( u i ) i ≥ be a sequence of non-decreasing positive integers. A (vector) ⃗ u -parking function of length n is a sequence ( x , ..., x n ) of positive integers whose order statistics, i.e., the nondecreasingrearrangement ( x ( ) , x ( ) , ..., x ( n ) ) , satisfy the inequalities x ( i ) ≤ u i for all i = , . . . , n . Vectorparking functions can be described via a parking process of n cars trying to park along a lineof x ≥ n parking spots. Cars enter one by one in order, and before parking, each driver has apreferred parking spot. Each driver goes to her preferred spot directly and parks in the firstspot available from there, if there exists one. A ⃗ u -parking function is a sequence of drivers’preferences such that at least i cars prefer to park in the first u i spots, for all i = , . . . , n . When u i = i , we recover the classical parking functions, which were originally introduced by Konheimand Weiss [7] and are the preference sequences such that x = n and every driver can find somespot to park. In general, the n -th Gonˇcarov polynomials associated to the pair ( D, −Z ) countsthe number of ⃗ z -parking functions of length n , where ⃗ z = ( z , z , . . . , z n − ) is the initial segmentof the grid Z ; see [9].A concrete realization of t n ( x ; ∆ , Z ) for some other delta operators ∆ is found in a combina-torial object called reluctant functions whose underlying structure are families of labeled trees.In [11], it is proved for some properly defined ∆ and Z , t n ( x ; ∆ , Z ) enumerates the number ofreluctant functions in a certain binomial class B whose label sequences are ⃗ z -parking functions.The object of the present paper is to extend this result and prove that for any delta operator, thegeneralized Gonˇcarov polynomials (up to a scaling) have a realization as a weighted enumeratorin partition lattices and in any exponential family. Gonˇcarov Polynomials in Partition Lattices
In this section we give a generic, or universal realization of generalized Gonˇcarov polynomials inweighted enumeration over partition lattices. Our result is built on Ray’s solution of the real-ization problem for arbitrary sequences of binomial types in the context of partition categories.Here, we will simplify his notation and present his construction in terms of incidence algebraof partially ordered sets, which was the language originally used by combinatorialists, e.g., seeJoni and Rota [6].For any finite set S , let Π ( S ) denote the set of all partitions of S , and write Π n for Π ([ n ]) .Elements of Π ( S ) are partially ordered by refinement: that is, define π ≤ σ if every block of π is contained in a block of σ . In particular, Π ( E ) has a unique maximal element ˆ1 that hasonly one block and a unique minimal element ˆ0 for which every block is a singleton. Let ∣ π ∣ bethe number of blocks of π and Π ( π ) be the partitions of the set that consists of blocks of π ,When π ≤ σ , the induced partition σ / π is the partition σ viewed as an element of Π ( π ) . Definethe class of ( π, σ ) as the sequence λ = ( λ , λ , ... ) of non-negative integers such that λ i is thenumber of blocks of size i in the partition σ / π , for 1 ≤ i ≤ ∣ π ∣ . It follows that ∑ i ≥ iλ i = ∣ π ∣ and ∑ i ≥ λ i = ∣ σ ∣ . Example 1.
Let E = [ ] , π = { } , { } , { } , { } , { } , σ = { } , { } , { } ∈ Π . Then, σ / π = {( ) , ( )} , {( )} , {( ) , ( )} ∈ Π ( π ) . The class of ( π, σ ) is λ = ( , , , , ... ) , where wehave ∑ i ≥ iλ i = ∣ π ∣ = ∑ i ≥ λ i = ∣ σ ∣ = P be a finite poset and A a commutativering with unity. Denote by Int ( P ) the set of all intervals of P , i.e., the set {( x, y ) ∶ x ≤ y } . The incidence algebra I ( P, A ) of P over A is the A -algebra of all functions f ∶ Int ( P ) → A, where multiplication is defined via the convolution f g ( x, y ) = ∑ x ≤ z ≤ y f ( x, z ) g ( z, y ) . The algebra I ( P, A ) is associative with identity δ , where δ ( x, y ) = ⎧⎪⎪⎨⎪⎪⎩ , if x = y, , if x ≠ y. An element f ∈ I ( P, A ) is invertible under the multiplication if and only if f ( x, x ) is invertiblein A for every x ∈ P .In this paper we are concerned with the case P = Π n , the partition lattice of [ n ] , and A = K [ w , w , . . . ] , where w , w , ... are independent variables. In addition, we set w = Definition 5.
Assume π ≤ σ in Π n and the class of ( π, σ ) is λ = ( λ , λ , . . . ) . Define the zeta-typefunction w ( π, σ ) ∈ I ( Π n , A ) by letting w ( π, σ ) = w λ w λ . . . w λ ∣ π ∣ ∣ π ∣ . (5) ote that w ( π, π ) = π . Hence w is invertible. The inverse of w is called the M¨obius-type function and denoted by µ w . Explicitly, µ w ( π, π ) = π < σ , µ w ( π, σ ) = − ∑ π ≤ τ < σ µ w ( π, τ ) w ( τ, σ ) . When all w i =
1, the zeta-type function and the M¨obius-type function become the zeta functionand the M¨obius function of Π n respectively. Example 2.
Consider the lattice Π . Then for all π < σ , w ( π, σ ) = w except that w ( ˆ0 , ˆ1 ) = w .Consequently, µ w ( π, σ ) = − w if π < σ except that µ w ( ˆ0 , ˆ1 ) = w − w .Define the zeta-type enumerator { a n ( x ; w )} n ≥ and M¨obius-type enumerator { b n ( x ; w )} n ≥ as follows. Let a ( x ; w ) = b ( x ; w ) = n ≥ a n ( x ; w ) = ∑ π ∈ Π n w ( ˆ0 , π ) x ∣ π ∣ , (6) b n ( x ; w ) = ∑ π ∈ Π n µ w ( ˆ0 , π ) x ∣ π ∣ . (7) Theorem 2 ([14]) .
1. The polynomial sequences { a n ( x ; w )} n ≥ and { b n ( x ; w )} n ≥ are of bi-nomial type.2. Let Λ be the delta operator whose D -indicator is given by g ( t ) = t + ∑ i ≥ w i t i / i ! . Then { a n ( x ; w )} n ≥ is the conjugate sequence of Λ and { b n ( x ; w )} n ≥ is the basic sequence of Λ . For n = , , . . . ,
4, the polynomials a n ( w, x ) and b n ( w, x ) are a ( x ; w ) = ,a ( x ; w ) = x,a ( x ; w ) = x + w x,a ( x ; w ) = x + w x + w x,a ( x ; w ) = x + w x + ( w + w ) x + w x, and b ( x ; w ) = ,b ( x ; w ) = x,b ( x ; w ) = x − w x,b ( x ; w ) = x − w x + ( w − w ) x,b ( x ; w ) = x − w x + ( w − w ) x + ( w w − w − w ) x. The linear coefficient in b n ( w ; x ) is µ wn = µ w ( ˆ0 , ˆ1 ) in Π n . Assume ∆ is the conjugate deltaoperator of Λ. Then { a n ( x ; w )} n ≥ is the basic sequence of ∆ and { b n ( x ; w )} n ≥ is the conjugatesequence of ∆. The operator ∆ can be written as ∆ = ∑ n ≥ µ wn D n / n !. Since w =
1, each µ wn is a polynomial of w , w , . . . . If we take w to be a variable, µ wn would be a polynomial in w − , w , w , . . . .The condition w = a ( x ; w ) = x . Since the weight vari-ables w , w , . . . can take arbitrary values, Theorem 2 implies that any polynomial sequence p n ( x )} n ≥ of binomial type with p ( x ) = x can be realized as the zeta-type weight enumeratoror the M¨obius-type weight enumerator over partition lattices. Note that for any scalar k ≠ { p n ( x )} n ≥ is the basic sequence of ∆ and the conjugate sequence of Λ, then { p n / k n } n ≥ is the basic sequence of k ∆ and the conjugate sequence of g ( D / k ) where g ( t ) is the D -indicator of Λ. Hence Theorem 2 covers all polynomial sequences of binomial type up to ascaling.In the problem of counting assemblies of B -structures outlined in Section 2.1, the enumerator ∑ k b n,k x k in Theorem 1 is a specialization of the polynomial a n ( x ; w ) , where w n is the numberof B -structures on a block of size n . For example, when B is the set of rooted trees, w n = n n − and hence a n ( x ; w ) = x ( x + n ) n − , the n -th Abel polynomial.Our objective is to fit the generalized Gonˇcarov polynomials into this model and presenta combinatorial interpretation in terms of weight-enumeration in partition lattices. Followingthe notation of Theorem 2, let ∆ be the conjugate delta operator of Λ. Given an interpolationgrid Z , we denote by t n ( x ; w, Z ) the n -th generalized Gonˇcarov polynomial relative to the pair ( ∆ , Z ) . We use this notation to emphasize the role of the zeta-type function w ( π, σ ) .To get a formula for the polynomial t n ( x ; w, Z ) , we use the recurrence (4) in Section 2.2.Since a n ( x ; w ) is the basic sequence of ∆, { t n ( x ; w, Z )} n ≥ is the unique sequence of polynomialsthat satisfies the recurrence a n ( x ; w ) = n ∑ i = ( ni ) a n − i ( z i ; w ) t i ( x ; w, Z ) . (8)In other words, t n ( x ; w, Z ) = a n ( x ; w ) − n − ∑ i = ( ni ) a n − i ( z i ; w ) t i ( x ; w, Z ) . (9)In particular, t ( x ; w, Z ) = t ( x ; w, Z ) = a ( x ; w ) − a ( z ; w ) = x − z . Here we again assume w = a ( x ; w ) = x . Since if ∆ is changed to k ∆, the corresponding t n ( x ; w, Z ) justchanges to t n ( x ; w, Z )/ k n , again we cover all the cases up to a scaling.Assume x is a positive integer and X = { , , . . . , x } . Then a n ( x ; w ) is the zeta-type weightenumerator of all the block-labeled partitions, where each block of the partition carries a labelfrom X . In symbols, a n ( x ; w ) = ∑ π ∈ Π n w ( ˆ0 , π ) ⋅ ∣{ f ∶ Block ( π ) → X }∣ , where Block ( π ) is the set of blocks of π . For a partition π with a block-labeling f , we recordthe labeling by the list f π = ( x , x , . . . , x n ) , where x i = f ( B j ) whenever i is in the block B j of π . Let ⃗ z = ( z , z , ⋯ , z n − ) be the initial segment of the grid Z . Furthermore, assume that z ≤ z ⋯ ≤ z n − are positive integers with z n − < x .Define the set PF π ( Z ) as the set of all block-labelings of π that are also ⃗ z -parking functions,i.e., PF π ( Z ) = { f ∶ Block ( π ) → X ∣ f π is a ⃗ z -parking function } . (10)More precisely, PF π ( Z ) is the set of block-labelings of π such that the order statistics of f π = ( x , x , . . . , x n ) satisfies x ( i ) ≤ z i − for i = , . . . , n . Let P F π ( Z ) be the cardinality of PF π ( Z ) . ur main result of this section is the following theorem. Theorem 3.
Assume t n ( x ; w, Z ) is the n -th generalized Gonˇcarov polynomial defined by (8) with a positive increasing integer sequence Z = ( z , z , ... ) . Let x be an integer larger than z n − .Then, t n ( ω, − Z ) = t n ( x ; ω, x − Z ) = ∑ π ∈ Π n w ( ˆ0 , π ) ⋅ P F π ( Z ) , (11) where x − Z = ( x − z , x − z , x − z , . . . ) and − Z = ( − z , − z , − z , . . . ) . The first equality follows from [11, Prop.3.5] that was proved by verifying the defining equa-tion (3), and the second equality follows from the recurrence (8) and Lemma 4 proved next.Note that all three parts of (11) are polynomials of z , z , . . . , z n − , hence (11) is a polynomialidentity. Lemma 4.
For every n ≥ , it holds that a n ( x ; w ) = n ∑ i = ( ni ) a n − i ( x − z i ; w ) ∑ π ∈ Π i w ( ˆ0 , π ) ⋅ P F π ( Z ) (12) Proof.
Again we assume that x and z i are positive integers and z < z < ⋯ < z n − < x . For afinite set E and P , let S ( E, P ) be the set of pairs ( π, f ) where π is a partition of the set E and f is a function from Block ( π ) to P . Then the left-hand side of (12) counts the set S ([ n ] , X ) by the zeta-type weight function w ( ˆ0 , π ) . Note that if π has blocks B , B , . . . , B k , then w ( ˆ0 , π ) = k ∏ j = w ∣ B j ∣ . For a pair ( π, f ) ∈ S ([ n ] , X ) with f π = ( x , x , . . . , x n ) , let inc ( f π ) = ( x ( ) , x ( ) , . . . , x ( n ) ) bethe non-decreasing rearrangement of the terms of f π . Set i ( f ) = max { k ∶ x ( j ) ≤ z j − ∀ j ≤ k } . Thus, the maximality of i = i ( f ) means that x ( ) ≤ z , x ( ) ≤ z , ... , x ( i ) ≤ z i − and z i < x ( i + ) ≤ x ( i + ) ≤ ⋯ ≤ x ( n ) ≤ x. In the case that x ( j ) > z j − for all j , we have i ( f ) = ( x r , ..., x r i ) is the subsequence of f π from which the non-decreasing sequence ( x ( ) , x ( ) , ..., x ( i ) ) is obtained. Let R = { r , r , . . . , r i } ⊆ [ n ] . Then it is easy to see that R must be a union of some blocks of π , while R = [ n ] ∖ R is the union of the remaining blocksof π . Let π be the restriction of π on R and π the restriction of π on R . Thus π is a disjointunion of π and π . Furthermore, let f i be the restriction of f on R i . Then f is a map fromthe blocks of π to { , . . . , z i } that is also a ⃗ z -parking function, and f is a map from blocks of π to the set X ∖ [ z i ] = { z i + , . . . , x } .Let S P ( E, X ) be the subset of S ( E, X ) such that for each pair ( π, f ) , the sequence f π isa ⃗ z -parking function. Then the above argument defines a decomposition of ( π, f ) ∈ S ([ n ] , X ) nto pairs ( π , f ) ∈ S P ( R , X ) and ( π , f ) ∈ S ( R , X ∖ [ z i ]) . Conversely, any pairs of ( π , f ) and ( π , f ) described above can be reassembled into a partition π of [ n ] with labels in X . Inother words, the set S ([ n ] , X ) can be written as a disjoint union of Cartesian products as S ([ n ] , X ) = ⊍ i ; R ⊆ [ n ] ∶ ∣ R ∣ = i S P ( R , X ) × S ( R , X ∖ [ z i ]) . (13)In addition, if π is the disjoint union of π and π , then w ( ˆ0 , π ) = w ( ˆ0 , π ) w ( ˆ0 , π ) . Putting the above results together, we have a n ( x ; w ) = ∑ ( π,f ) ∈ S ([ n ] ,X ) w ( ˆ0 , π ) = n ∑ i = ∑ R ∶ ∣ R ∣ = i ⎛⎝ ∑ ( π ,f ) ∈ S p ( R ,X ) w ( ˆ0 , π ) ⋅ ∑ ( π ,f ) ∈ S ( R ,X ∖ [ z i ]) w ( ˆ0 , π )⎞⎠ = n ∑ i = ( ni ) a n − i ( x − z i ; w ) ∑ ( π ,f ) ∈ S p ( R ,X ) w ( ˆ0 , π ) = n ∑ i = ( ni ) a n − i ( x − z i ; w ) ∑ π ∈ Π i w ( ˆ0 , π ) P F π ( Z ) . The last equation follows from the definition of
P F π ( Z ) . Example 3.
From the recurrence (8) we get t ( x ; w, Z ) = x + ( w − z ) x + ( z z − z − w z ) . Hence t ( w, − Z ) = z z − z + w z . On the other hand, there are two partitions in Π . For π = { } , clearly w ( ˆ0 , { }) = w and P F { } ( Z ) = z . For π = { }{ } , w ( ˆ0 , π ) = P F π ( Z ) is the number of pairs of positive integers ( x, y ) such that min ( x, y ) ≤ z and max ( x, y ) ≤ z . Itis easy to check that there are 2 z z − z such pairs.Since { a n ( x ; w )} gives a generic form of the sequence of polynomials of binomial type, { t n ( x ; w, Z )} is the generic form of the generalized Gonˇcarov polynomials. In particular, fromTheorem 3 we see that when w = w = ⋯ = t n ( w, − Z ) gives the number of ⃗ z -parkingfunctions of length n . In this section, we explore a more concrete realization of generalized Gonˇcarov polynomials inexponential families, which are picturesque models that deal with counting structures that arebuilt out of connected pieces and can be applied to many combinatorial problems.
Exponential families are combinatorial models based on the partition lattices where the enu-meration are captured by the exponential generating functions. The description of exponential amilies and their relation to the incidence algebra of Π n can be found in standard textbooks,e.g., [16, Section 5.1]. Here we adopt Wilf’s description of exponential families [18] in the contextof ‘playing cards’ and ‘hands’.Suppose that there is given an abstract set P of ‘pictures’, which typically are the connectedstructures. A card C ( S, p ) is a pair consisting of a finite label set S of positive integers and apicture p ∈ P . The weight of C is ∣ S ∣ . If S = [ n ] , the card is called standard . A hand H is aset of cards whose label sets form a partition of [ n ] for some n . The weight of a hand is thesum of the weights of the cards in the hand. The n -th deck D n is the set of all standard cardsof weights n . We require that D n is always finite. An exponential family F is the collection ofdecks D , D , . . . .In an exponential family, let d i = ∣ D i ∣ and h n,k be the number of hands H of weight n thatconsist of k cards. Let h ( x ) = n ≥ h n ( x ) = n ∑ k = h n,k x k . (14)Then the main counting theorem, the exponential formula , states that these polynomials satisfythe generating relation: e xD ( t ) = ∑ n ≥ h n ( x ) t n n ! , (15)where D ( t ) = ∑ k ≥ d k t k / k !. In other words, if d = h , ≠ { h n ( x )} n ≥ is a sequence of binomialtype that is conjugate to the delta operator ∑ k ≥ d k D k / k !. Example 4.
Set Partitions : Here, a card is a label set [ n ] with a ‘picture’ of n dots. Eachdeck D n consists of the single card of weight n , and a hand is just a partition of the set [ n ] .Thus, h n,k is the number of partitions of the set [ n ] into k classes, which is S ( n, k ) , the Stirlingnumber of the second kind. Example 5.
Permutations and their Cycles : Each card is a cyclic permutation on a label set S . The deck D n consists of all distinct cyclic permutations on [ n ] so d n = ( n − ) ! and a handis a permutation of [ n ] consisting of k cycles. Thus, h n,k is the number of permutations on [ n ] that have k cycles, that is, the signless Stirling number of the first kind c ( n, k ) .Note that we can interpret x k in h n ( x ) as the number of maps from the set of cards in ahand to the set X = { , . . . , x } for some positive integer x . Hence h n ( x ) counts the number ofhands of weight n in which each card is labeled by an element of X . This set-up gives a naturalcombinatorial interpretation for binomial polynomial sequences whose coefficients are positiveintegers.In Foundation III [12] Mullin and Rota introduced a structure called reluctant functions ,which can be used to give a combinatorial interpretation for some generalized Gonˇcarov poly-nomials; see [11]. We remark that reluctant functions is a special case of exponential families,in which the ‘pictures’ are certain sets of trees. We will show that by taking the type enumera-tor an exponential family actually provides a combinatorial model for all generalized Gonˇcarovpolynomials. .2 Type Enumerator in Exponential Families In a given exponential family F , we have seen that h n ( x ) = n ∑ k = h n,k x k = ∑ H ∣{ f ∶ cards in H → X }∣ (16)where H ranges over all hands of weight n . For a hand H consisting of cards C , C , . . . , C k ofweights t , t , . . . , t k , define the type of H astype ( H ) = y t y t ⋯ y t k , where y , y , . . . , are free variables.Let h n ( x ; y ) = ∑ H ∶ weight n type ( H )∣{ f ∶ cards in H → X }∣ . (17)Then we have the following form of the exponential formula. Proposition 5.
The sequence of type enumerators { h n ( x ; y )} n ≥ , viewed as a polynomial in x ,is a sequence of polynomials of binomial type satisfying the equation ∑ n ≥ h n ( x ; y ) t n n ! = exp ( x ∑ k ≥ d k y k t k k ! ) . (18) Proof.
We compare the formula of h n ( x ; y ) with that of h n ( x ) . Note for n ≥ h n ( x ) can becomputed by h n ( x ) = ∑ k ≥ ∑ H = { C ,..., C k } d t d t ⋯ d t k x k , (19)where { C , ⋯ , C k } is a hand of weight n and t i is the weight of card C i , while h n ( x ; y ) = ∑ k ≥ ∑ H = { C ,..., C k } d t d t ⋯ d t k y t y t ⋯ y t k x k . (20)The exponential formula for h n ( x ) then implies Proposition 5. Remark.
Comparing to the generic form a n ( x ; w ) in the previous section, we see that h n ( x ; y ) corresponds to the case where the variables in the zeta-type function are determined by w n = d n y n . As far as d ≠
0, we can obtain arbitrary polynomial sequences of binomial type by takingsuitable values for the y i -variables. Let Z = ( z i ) i ≥ be an interpolation grid. For the binomial sequence { h n ( x ; y )} n ≥ defined in anexponential family F , we can consider the associated generalized Gonˇcarov polynomials givenby (4) with p n ( x ) replaced by h n ( x ; y ) . Denote this Gonˇcarov polynomial by t n ( x ; y , F , Z ) toemphasize that it has variables y i and is defined in F . Explicitly, t n ( x ; y , F , Z ) is obtained bythe recurrence t n ( x ; y , F , Z ) = h n ( x ; y ) − n − ∑ i = ( ni ) h n − i ( z i ; y ) t i ( x ; y , F , Z ) . (21) uppose X = { , , ..., x } and assume that z ≤ z ≤ ⋯ ≤ z n − are integers in X . Let ⃗ z = ( z , z , . . . , z n − ) . For a hand H = { C , C , . . . , C k } of weight n with a function f from { C , C , . . . , C k } to X , denote by f H the list ( x , x , . . . , x n ) , where x i = f ( C j ) if i is in the labelset of C j . Let PF H (Z) = { f ∶ {C , C , . . . , C k } → X ∣ f H is a ⃗ z -parking function } , and P F H (Z) the cardinality of PF H (Z) . Then we have the following analog of Theorem 3. Theorem 6.
For n ≥ , t n ( y , F , − Z) = t n ( x ; y , F , x − Z) = ∑ H ∶ of weight n type ( H ) ⋅ P F H (Z) . (22) Here by convention, the third term of (22) equals when n = . Theorem 6 follows from (21) and the following recurrence relation h n ( x ; y ) = n ∑ i = ( ni ) h n − i ( x − z i ; y ) ∑ H ∶ of weight i type ( H ) ⋅ P F H (Z) , (23)whose proof is similar to that of Lemma 4. In an exponential family F , let A( S, X ) be the setof pairs ( H, f ) such that H is a hand whose label sets form a partition of S and f is a functionfrom the cards in H to X . Then the basic ingredients of the proof are that(1) type ( H ) is a multiplicative function only depending on the weights of cards in H , and(2) The set A([ n ] , X ) can be decomposed into a disjoint union of Cartesian products of theform A P ( R, X ) × A([ n ] ∖ R, X ∖ [ z i ]) , where A P ( R, X ) = {( H, f ) ∈ A( R, X ) ∶ f H is a ⃗ z -parking function } , and the disjoint unionis taken over all the subsets R of [ n ] .We skip the details of the proof of Eq. (23).We illustrate the above results and some connections to combinatorics in the exponentialfamilies given in Examples 4 and 5. There are many other exponential families in which the typeenumerator and associated Gonˇcarov polynomials have interesting combinatorial significance.1. Let F be the exponential family of set partitions described in Example 4. In this family, d i = i and h n ( x ) = n ∑ k = S ( n, k ) x k . In the type enumerator, if we substitute y = y i = w i for i ≥
2, then h n ( x ; y ) is exactly the same as the generic sequence a n ( x ; w ) in (6), and consequently t n ( x ; y , F , Z) is the same as the generic Gonˇcarov polynomial t n ( x ; w, Z) defined by (9). In particular, if all y i = t n ( y , F , − Z) gives a formula forthe number of ⃗ z -parking functions with the additional structure that cars arrive in disjointgroups, and drivers in the same group always prefer the same parking spot.When y i = z i = + i for all i , the first few terms of the Gonˇcarov polynomials n ( x ) = t n ( x ; y , F , − Z) are t ( x ) = t ( x ) = x + t ( x ) = x + x + t ( x ) = x + x + x + t ( x ) = x + x + x + x + x = , , , , , ... . This is sequence A030019 inthe On-Line Encyclopedia of Integer Sequences (OEIS) [13], where it is interpreted as thenumber of labeled spanning trees in the complete hypergraph on n vertices (all hyper-edgeshaving cardinality 2 or greater). It would be interesting to find a direct bijection betweenthe hyper-trees and the parking-function interpretation.2. Let F be the exponential family of the permutations and their cycles, as described inExample 5. Here d n = ( n − ) ! and h n ( x ) = n ∑ k = c ( n, k ) x k = x ( n ) , where the c ( n, k ) is thesignless Stirling numbers of the first kind and x ( n ) is the rising factorial x ( x + ) ⋯ ( x + n − ) .When y =
1, the Gonˇcarov polynomial t n ( x ; y , F , Z) can be obtained from the genericform t n ( x ; w, Z) by replacing w n with ( n − ) ! y n for n ≥
2. When all y i =
1, i.e. y = , t n ( , F , − Z) gives a formula for the number of ⃗ z -parking functions with the additionrequirement that cars are formed in disjoint cycles, and drivers in the same cycle preferthe same parking spot.In addition, when y = , and Z is the arithmetic progression z i = a + bi , the Goncarovpolynomial is t n ( x ; , F ; − Z) = ( x + a )( x + a + nb + ) ( n − ) . (24)Another combinatorial interpretation of t n ( , F , − Z) is given in [11, Section 6.7], whereit shows that t n ( , F , − Z) is n ! times the number of lattice paths from ( , ) to ( x − , n ) with strict right boundary Z . For example, when z i = a + bi for some positive integers a and b , n ! t n ( , F , − Z) is the number of lattice paths from ( , ) to ( x − , n ) which staystrictly to the left of the points ( a + ib, i ) for i = , , . . . , n . In particular for a = b = k , it counts the number of labeled lattice paths from the origin to ( kn, n ) that neverpass below the line x = yk . In that case (24) gives + kn ( ( k + ) nn ) , the n -th k -Fuss-Catalannumber. Remark.
We can also consider the injective functions in the definition of h n ( x ) and h n ( x ; y ) in(16) and (17), where the term x k is replaced by the lower factorial x ( k ) = x ( x − ) ⋯ ( x − k + ) . Inother words, cards of a hand are labeled by X with the additional property that different cardsget different labels. Some examples are given in [11, Section 6] and called monomorphic classes .A result analogous to Theorem 6 still holds for the momomorphic classes of an exponentialfamily.As a final result we point out an explicit formula to compute the constant coefficient of thegeneralized Gonˇcarov polynomial whenever we know the basic sequence { p n ( x )} n ≥ . It is provedin [11] and only depends on the recurrence (4) and the fact that p n ( ) = n >
0. The proof oes not need an explicit formula for the delta operator ∆ and hence the result is easier to usewhen we need to compute the value of t n ( y , F , − Z) in a given exponential family.Let { p n ( x )} n ≥ be a sequence of binomial type and Z = ( z , z , . . . ) be a given grid. Assume { t n ( − Z)} n ≥ is defined by the recurrence relation t n ( − Z) = − n − ∑ i = ( ni ) p n − i ( − z i ) t i ( ) , for n ≥ t ( − Z) =
1. Then for n ≥ t n ( − Z) can be expressed as a summation overordered partitions.Given a finite set S with n elements, an ordered partition of S is an ordered list ( B , ..., B k ) of disjoint nonempty subsets of S such that B ∪ ⋯ ∪ B k = S . If ρ = ( B , ..., B k ) is an orderedpartition of S , then we set ∣ ρ ∣ = k . For each i = , , ..., k , we let b i = b i ( ρ ) = ∣ B i ∣ , and s i ∶ = s i ( ρ ) ∶ = ∑ ij = b j . In particular, set s ( p ) =
0. Let R n be the set of all ordered partitions of the set [ n ] . Theorem 7 ([11]) . For n ≥ , t n ( − Z) = ∑ ρ ∈ R n ( − ) ∣ ρ ∣ k − ∏ i = p b i + ( − z s i ) = ∑ ρ ∈ R n ( − ) ∣ ρ ∣ p b ( − z ) , ⋯ p b k ( − z s k − ) . (25)The following list is the formulas for the first several Gonˇcarov polynomials. t ( − Z) = t ( − Z) = − p ( − z ) t ( − Z) = p ( − z ) p ( − z ) − p ( − z ) t ( − Z) = − p ( − z ) + p ( − z ) p ( − z ) + p ( − z ) p ( − z ) − p ( − z ) p ( − z ) p ( − z ) . In an exponential family, the polynomial h n ( x ) or h n ( x ; y ) may not always have degree n , e.g.,when d = h , =
0. We say that such polynomial sequences and the corresponding exponentialfamilies are degenerate . For a degenerate sequence of polynomials, there is no delta operatorfor which the sequence is the basic or the conjugate sequence. Nevertheless, the exponentialformulas (15) and (18) are still true. Hence the sequences { h n ( x )} n ≥ and { h n ( x ; y )} n ≥ stillsatisfy the binomial-type identity (1).Without a delta operator, we cannot define the generalized Gonˇcarov interpolation problems.However, we can still introduce the generalized Gonˇcarov polynomials via the recurrence (4).Furthermore, we will prove in Theorem 8 that the shift invariance of Gonˇcarov polynomialscan also be derived from (4). Therefore, Theorems 6 and 7 still hold true for the degenerateexponential families since all the proofs follow from the binomial-type identity (1) and therecurrence (4). Theorem 8.
Assume { p n ( x )} n ≥ is a polynomial sequence of binomial type with p ( x ) = , butthe degree of a n ( x ) is not necessary n . Let t n ( x ; Z) be defined by the recurrence relation t n ( x ; Z) = p n ( x ) − n − ∑ i = ( ni ) p n − i ( z i ) t i ( x ; Z) . (26) or any scalar η and the interpolation grid Z = { z , z , z , . . . ) , let Z + η be the sequence ( z + η, z + η, z + η, ⋯ ) . Then we have t n ( x + η ; Z + η ) = t n ( x ; Z) (27) for all n ≥ .Proof. We prove Theorem 8 by induction on n . The initial case n = t ( x ; Z) = x and any grid Z . Assume Eq. (27) is true for all indices less than n . We compute t n ( x + η ; Z + η ) . By definition t n ( x + η ; Z + η ) = p n ( x + η ) − n − ∑ i = ( ni ) p n − i ( z i + η ) t i ( x + η ; Z + η ) . (28)By the inductive hypothesis t i ( x + η ; Z + η ) = t i ( x ; Z) for i < n and the binomial identity of p n ( x ) , the right-hand side of (28) can be written as n ∑ k = ( nk ) p k ( x ) p n − k ( η ) − n − ∑ i = ( ni ) ⎛⎝ n − i ∑ j = ( n − ij ) p n − i − j ( z i ) p j ( η )⎞⎠ t i ( x ; Z) = n ∑ k = ( nk ) p k ( x ) p n − k ( η ) − ∑ i + j ≤ n except ( i,j )=( n, ) ( ni )( n − ij ) p j ( η ) p n − i − j ( z i ) t i ( x ; Z) (29)Since ( ni )( n − ij ) = n ! i ! j ! ( n − i − j ) ! = ( nj )( n − ji ) , then (29) can be expressed as n ∑ k = ( nk ) p k ( x ) p n − k ( η ) − ∑ i + j ≤ n except ( i,j )=( n, ) ( nj )( n − ji ) p j ( η ) p n − i − j ( z i ) t i ( x ; Z) = n ∑ k = ( nk ) p k ( x ) p n − k ( η ) − n ∑ j = ( nj ) p j ( η ) n − j ∑ i = ( n − ji ) p n − i − j ( z i ) t i ( x ; Z) − n − ∑ i = ( ni ) p n − i ( z i ) t i ( x ; Z) . (30)The last summation in (30) corresponds to the terms with j =
0. Note that n − j ∑ i = ( n − ji ) p n − i − j ( z i ) t i ( x ; Z) = p n − j ( x ) . Hence For. (30) is equal to n ∑ k = ( nk ) p k ( x ) p n − k ( η ) − n ∑ j = ( nj ) p j ( η ) p n − j ( x ) − n − ∑ i = ( ni ) p n − i ( z i ) t i ( x ; Z) = p n ( x ) − n − ∑ i = ( ni ) p n − i ( z i ) t i ( x ; Z) = t n ( x ; Z) . This finishes the proof. he next example shows a degenerate exponential family. Example 6. . In this exponential family a card is an undirected cycleon a label set [ m ] (where m ≥ D n consists of all undirected circular arrangementsof n letters so d n = ( n − ) ! for n ≥ d = d =
0. A hand is then a undirected simple graphon the vertex set [ n ] , which is 2-regular, that is, every vertex has degree 2. Thus, h n,k is thenumber of undirected 2-regular simple graphs on n vertices consisting of k cycles. Denote by F this exponential family.For F , the type enumerators are h ( x, y ) = h ( x ; y ) = h ( x ; y ) = h ( x ; y ) = y x , h ( x ; y ) = y x , h ( x, y ) = y x , and h ( x ; y ) = y x + y x , etc. Although the degree of h n ( x ; y ) is not n , the exponential formula still holds: n ∑ k = h n ( x ; y ) t k k ! = exp ( x ∑ k ≥ y k t k k ) . We compute by the recurrence (21) that t ( x ; y , F , Z) = t ( x ; y , F , Z) = t ( x ; y, F , Z) = t ( x ; y , F , Z) = y ( x − z ) ,t ( x ; y , F , Z) = y ( x − z ) ,t ( x ; y , F , Z) = y ( x − z ) ,t ( x ; y , F , Z) = y x + y x − y z x − y z − y z + y z z . The equation t n ( y , F , − Z) = ∑ H ∶ of weight n type ( H ) ⋅ P F H (Z) is still true. For example, for n = t ( y, F , − Z) = y z + y z z − y z . The term60 y z comes from the 5! / =
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