Lattice Path Enumeration and Its Applications in Representation Theory
aa r X i v : . [ m a t h . C O ] A ug Lattice Path Enumeration and Its Applications inRepresentation Theory
Jianqiang Feng Wenli Liu Ximei Bai Zhenheng Li
Abstract
In this paper, we enumerate lattice paths with certain constraints and apply thecorresponding results to develop formulas for calculating the dimensions of submod-ules of a class of modules for planar upper triangular rook monoids. In particular,we show that the famous Catalan numbers appear as the dimensions of some specialmodules; we also obtain some combinatorial identities.
Keywords:
Lattice path, module, dimension, Catalan number, rook monoid, orderpreserving and order decreasing.
In this paper we find an application of monotonic lattice paths in the representationtheory of a planar upper triangular rook monoid IC n consisting of order preserving andorder decreasing partial maps of n = { , ..., n } . The modules of interest are submodulesof V = L nk =0 V k , where V k is a vector space over a field F of characteristic zero generatedby a set of elements v S indexed by the k -subsets of n , with the module structure underthe action: for f ∈ IC n and S ⊆ n , f · v S = (cid:26) v S ′ , if S ⊆ D ( f )0 , otherwise,where D ( f ) is the domain of f , S ′ = f ( S ). We use monotonic lattice path enumerationto describe the dimensions of the submodules of V .Lattice paths under certain constraints have been studied in combinatorics over along period and many elegant results have emerged (see for example [5, 10, 11]), withapplications to problems in probability and statistics: the traditional gambler’s ruin, therank order statistics for non-parametric testing [12, 18], and distributional problems inrandom walks [16]. Recently researchers find applications of lattice paths in commu-tative algebra; non-intersecting lattice paths are used to describe the Hilbert series ofdeterminantal and Pfaffian rings [8, 15].The monoids we treat are closely related to the theory of linear algebraic monoids;we are here dealing with a submonoid of the most familiar interesting case of the Rennermonoids of reductive monoids [20, Scetion 8.5]. For more information on the Rennermonoids, see [19, 20, 21, 22, 14, 3]. 1he organization of the paper is as follows. In Section 2 we gather some necessaryconcepts and basic facts about lattice path enumeration and the planar upper triangularrook monoid.In Section 3 for a given decreasing lattice path from the point (0 , λ ) to the point( k,
0) with the height sequence λ ≥ · · · ≥ λ k , we develop an iterative formula in Theorem3.2 for enumerating decreasing lattice paths from (0 , λ ) to ( k,
0) and below the givenpath. This formula is different from the existing ones for such enumeration, and to ourknowledge it seems new. As consequences, we obtain two combinatorial identities inCorollaries 3.4 and 3.5, of which the latter is related to the Catalan numbers.In Section 4 we provide three formulas for computing the dimensions of submodulesof V via lattice path enumeration. The first formula stated in Theorem 4.6 calculates thedimensions of the submodules h v S i of V generated by a single basis vector v S where S is asubset of n ; this formula is an application of Theorem 3.2. As a result of the application of[11, Theorem 10.7.1], the second formula for computing the dimension of the submodule h v S i is given in Theorem 4.11. It turns out that the famous Catalan numbers appearas the dimensions of some submodules h v S i . The third formula described in Theorem4.12 calculates the dimension of every submodule of V . For developing these formulaswe have investigated the properties of submodules of V , and in this paper we only showthe properties needed for establishing our formulas. We find that the submodule h v S i isa direct sum of the vector spaces F v T where T ranges over all subsets less that or equalto S . We then show that every submodule of V is cyclic and contains a unique reducedgenerator. Furthermore we conclude that two submodules equal if and only if they havethe same reduced generator. Acknowledgement
We would like to thank Dr. M. Can for useful email communi-cations and Dr. R. Koo for valuable comments.
We gather necessary definitions and basic facts about lattice paths and planar uppertriangular rook monoids.
A lattice path v is a sequence of finite lattice points v , . . . , v k of Z , with v thestarting point and v k the ending point of the path. The vectors −−→ v v , −−→ v v , . . . , −−−−→ v l − v k arereferred to as the steps of the path. A lattice path is decreasing if each step is either(1 ,
0) or (0 , − ,
0) or (0 , monotonic if it is decreasing or increasing. The lattice path from (0 ,
4) to(5 ,
0) in the left figure below is decreasing, and the one from (0 ,
0) to (5 ,
4) in the rightis increasing. 2 rrrt rrrtt rrrtr rtttr rtrrr ttrrr (0 ,
4) 4 3 3 11 (5 , ttrrr rtttr rrrtr rrrtt rrrrt rrrrt (0 ,
0) 1 3 3 4 4 (5 , x -axis, and the nonnegative y -axis. We further assume that the starting points of thepaths are all on the nonnegative y -axis.The heights λ i of all the horizontal steps of a monotonic lattice path form a finitesequence ( λ , . . . , λ k ) of nonnegative integers, which is uniquely determined by the latticepath, and is called the height sequence of the path. If the path is decreasing (resp.increasing) its height sequence is denoted by λ ≥ . . . ≥ λ k (resp. λ ≤ . . . ≤ λ k ). Theheight sequence of the path in the left figure above is 4 ≥ ≥ ≥ ≥
1, and that of thepath in the right is 1 ≤ ≤ ≤ ≤ λ ≥ . . . ≥ λ k of nonnegative integers, there existsa unique decreasing lattice path in the closed first quadrant from the point (0 , λ ) tothe point ( k, a ≤ . . . ≤ a k of nonnegative integers there exists a unique increasinglattice path in the closed first quadrant from the point (0 ,
0) to the point ( k, a k ), whoseheight sequence is the given sequence.It is convenient to identify a decreasing lattice path from the point (0 , λ ) to the point( k,
0) with its height sequence λ ≥ . . . ≥ λ k , and identify an increasing lattice path fromthe point (0 ,
0) to the point ( k, a k ) with its height sequence a ≤ . . . ≤ a k , and we oftendo so without mentioning it further. We refer the reader to [10, 11] and the referencescited there for a comprehensive survey of lattice path enumeration. An injective partial map f of n is a one-to-one map of a subset D ( f ) of n onto asubset R ( f ) of n where D ( f ) is the domain of f and R ( f ) is the range of f . We agreethat there is a map with empty domain and range and call it 0 map.We can represent an injective partial map by an n × n matrix, where the entry in the i th row and the j th column is 1 if the map takes j to i , and is 0 otherwise, such a matrixis named a rook matrix, a matrix with at most one 1 in each row and each column. Forexample, the map σ given below is an injective partial map of n = { , , , } , σ = (cid:18) × (cid:19) = . rook monoid R n is the monoid of injective partial maps from n to n , whoseoperation is the composition of partial maps and the identity element is the identitymap of n . Since elements of R n are not necessarily invertible, R n is not a group. Themap with empty domain and empty range behaves as a zero element. Identifying aninjective partial map with its associated rook matrix, R n can be regarded as the monoidconsisting of all the rook matrices of size n . The structures and representations of therook monoid are intensively studied [2, 7, 9, 17, 23]; the generating functions of R n andtheir connections to Laguerre polynomials are found in [1].We can write an injective partial map f of n in 2-line notation by writing the numbers s , . . . , s k in the top line if D ( f ) = { s , . . . , s k } , and then below each number we writeits image.An injective partial map from n to n is order preserving if whenever a < b in thedomain of the map, then f ( a ) < f ( b ). It is easily seen that an injective partial map f isorder preserving if and only if the graph obtained from the 2-line notation of f by joiningall defined f ( a ) in the range of the map to a is a planar graph, which justifies the namein the following definition. Definition 2.1.
The planar rook monoid is the monoid of order preserving injectivepartial maps from n to n . Obviously, a planar rook monoid is a submonoid of R n . The structure and represen-tation of the planar rook monoid is studied in [4].An injective partial map is called order decreasing if for all a in the domain of themap, we have f ( a ) ≤ a . Clearly, an injective partial map is order decreasing if and onlyif its matrix form is an upper triangular rook matrix, which motivates the name in thefollowing definition. Definition 2.2.
The planar upper triangular rook monoid , denoted by IC n , is the monoidof order preserving, order decreasing injective partial maps from n to n . The notation IC n for the planar upper triangular rook monoid is standard in semi-group theory, see for example [7, Chapter 14]. Let v be a decreasing lattice path in the closed first quadrant with the starting point(0 , λ ), the ending point ( k, λ ≥ . . . ≥ λ k . We say thata decreasing lattice path u with the height sequence µ ≥ . . . ≥ µ l is below v if it isfrom (0 , λ ) to ( k,
0) (hence l = k ) and 0 ≤ µ i ≤ λ i for i = 1 , . . . , k. In particular, if u is below v , then they share the same starting point and the same ending point. Theconcept ‘below’ for increasing lattice paths is defined similarly.The purpose of this section is to calculate the number d k of all the decreasing latticepaths below v . The next lemma is immediate.4 emma 3.1. The number d k is equal to the number of decreasing sequences µ ≥ · · · ≥ µ k of integers such that ≤ µ i ≤ λ i , i = 1 , . . . , k. We give a formula for calculating d k in Theorem 3.2. An example is useful to illustratethe idea of its proof. Let k = 2 and λ ≥ λ be 4 ≥
2. We first fix µ = 2 (= λ ); thereare 3 sequences µ ≥ µ with µ ≤ λ : 4 ≥
2; 3 ≥
2; 2 ≥
2. We then fix µ = 1; thereare 4 such sequences: 4 ≥
1; 3 ≥
1; 2 ≥
1; 1 ≥
1. We now fix µ = 0; there are 5 suchsequences : 4 ≥
0; 3 ≥
0; 2 ≥
0; 1 ≥
0; 0 ≥
0. Lemma 3.1 indicates d = 3 + 4 + 5 = 12 . From now on, we agree that if a > b , the empty sum P bi = a (cid:3) i = 0. Theorem 3.2.
Let v be a given decreasing lattice path in the closed first quadrant withthe starting point (0 , λ ) , the ending point ( k, , and the height sequence λ ≥ . . . ≥ λ k of nonnegative integers. Then the number d k of decreasing lattice paths below v is givenby d = λ + 1 , and for k ≥ by d k = k − X i =1 (cid:20)(cid:18) λ i + k − i + 1 k + 1 − i (cid:19) − (cid:18) λ i − λ k + k − ik + 1 − i (cid:19)(cid:21) γ i − k − X i =1 ( λ k + 1) (cid:18) λ i − λ k − + k − i − k − i (cid:19) γ i , where γ = 1 and γ j = − j − X i =1 (cid:18) λ i − λ j − + j − i − j − i (cid:19) γ i for j ≥ . (3.1) Proof.
To find d k , by Lemma 3.1 it suffices to compute the number of the sequences µ ≥ · · · ≥ µ k of nonnegative integers with µ i ≤ λ i for i = 1 , . . . , k. If k = 1, clearly d = λ + 1.If k ≥
2, let 2 ≤ j ≤ k . For each fixed nonnegative integer µ ≤ λ j , denote by α j ( µ )the number of sequences of nonnegative integers µ ≥ · · · ≥ µ j − ≥ µ with µ i ≤ λ i for i = 1 , . . . , j − . (3.2)We calculate α j ( µ ) iteratively on j , and the required number d k = P λ k µ =0 α k ( µ ).Let ξ j = λ j − µ . Then 0 ≤ ξ j ≤ λ j . Our aim now is to prove α j ( µ ) = β j + γ j , (3.3)where β j = P j − i =1 (cid:0) λ i − λ j + ξ j + j − ij − i (cid:1) γ i and γ j = − P j − i =1 (cid:0) λ i − λ j − + j − i − j − i (cid:1) γ i with γ = 1.Notice that α j ( µ ) is a sum of two numbers β j and γ j , of which γ j depends on λ , . . . , λ j − ,whereas β j depends on λ , . . . , λ j and ξ j .We use induction on j to prove (3.3) for 2 ≤ j ≤ k . If j = 2, for each fixed nonnegativeinteger µ ≤ λ we have ξ = λ − µ and 0 ≤ ξ ≤ λ . Let ξ = λ − µ . To ensure that(3.2) holds for this case, namely µ ≥ µ and µ ≤ λ , we must have 0 ≤ ξ ≤ λ − λ + ξ ,and conversely. So α ( µ ) = λ − λ + ξ + 1 = β + γ β = λ − λ + ξ + 1 and γ = 0, and this is (3.3) for j = 2.Suppose (3.3) holds for j = l with 2 ≤ l ≤ k −
1, that is, for each fixed nonnegativeinteger µ ≤ λ l we have ξ l = λ l − µ with 0 ≤ ξ l ≤ λ l , and the number of sequences µ ≥ · · · ≥ µ l − ≥ µ with µ i ≤ λ i for i = 1 , . . . , l − α l ( µ ) = β l + γ l , (3.4)where β l = P l − i =1 (cid:0) λ i − λ l + ξ l + l − il − i (cid:1) γ i and γ l = − P l − i =1 (cid:0) λ i − λ l − + l − i − l − i (cid:1) γ i .We now prove (3.3) for j = l + 1. For a fixed nonnegative integer ν ≤ λ l +1 we have ξ l +1 = λ l +1 − ν with 0 ≤ ξ l +1 ≤ λ l +1 . Let µ = λ l − ξ l . To ensure that the condition (3.2) µ ≥ · · · ≥ µ l − ≥ µ ≥ ν with µ i ≤ λ i , i = 1 , . . . , l − µ ≤ λ l holds here, we must have 0 ≤ ξ l ≤ ρ l where ρ l = λ l − λ l +1 + ξ l +1 , and conversely. Addingall α l ( µ ) up for ν ≤ µ ≤ λ l and using the induction hypothesis (3.4), we obtain α l +1 ( ν ) = λ l X µ = ν α l ( µ ) = λ l X µ = ν ( β l + γ l )= ρ l X ξ l =0 l − X i =1 (cid:18) λ i − λ l + ξ l + l − il − i (cid:19) γ i + ρ l X ξ l =0 γ l (3.5)= l − X i =1 (cid:26)(cid:18) λ i − λ l +1 + ξ l +1 + ( l + 1) − il + 1 − i (cid:19) γ i − (cid:18) λ i − λ l + l − il + 1 − i (cid:19) γ i (cid:27) + (cid:18) λ l − λ l +1 + ξ l +1 + 11 (cid:19) γ l (3.6)= l X i =1 (cid:18) λ i − λ l +1 + ξ l +1 + ( l + 1) − il + 1 − i (cid:19) γ i − l − X i =1 (cid:18) λ i − λ l + l − il + 1 − i (cid:19) γ i = β l +1 + γ l +1 , where β l +1 = l X i =1 (cid:18) λ i − λ l +1 + ξ l +1 + ( l + 1) − il + 1 − i (cid:19) γ i ,γ l +1 = − l − X i =1 (cid:18) λ i − λ l + l − il + 1 − i (cid:19) γ i . Here we have made use of the identity P a + b − z = a (cid:0) zp (cid:1) = (cid:0) a + bp +1 (cid:1) − (cid:0) ap +1 (cid:1) in which a, b, p are natural numbers to obtain (3.6) from (3.5) by assigning a = λ i − λ l + l − i, b = λ l − λ l +1 + ξ l +1 + 1 and p = l − i ≥
1. Therefore, (3.3) is valid for j = l + 1, and wecomplete the proof of (3.3) by induction. 6e are now able to calculate the number d k for k ≥ α k ( µ ) in (3.3)up where µ runs from 0 to λ k , yielding d k = λ k X µ =0 α k ( µ )= λ k X ξ k =0 k − X i =1 (cid:18) λ i − λ k + ξ k + k − ik − i (cid:19) γ i − λ k X ξ k =0 k − X i =1 (cid:18) λ i − λ k − + k − i − k − i (cid:19) γ i = k − X i =1 (cid:18) λ i + k − i + 1 k + 1 − i (cid:19) γ i − k − X i =1 (cid:18) λ i − λ k + k − ik + 1 − i (cid:19) γ i − k − X i =1 ( λ k + 1) (cid:18) λ i − λ k − + k − i − k − i (cid:19) γ i , which is the desired result.The next result is well-known, and is a special case of [11, Theorem 10.7.1] enumer-ating the number of increasing lattice paths below a given increasing lattice path. Notethat [11, Theorem 10.7.1] was initially obtained in [13] using recurrence relations. Recallthat for a given increasing lattice path v in the closed first quadrant from (0 ,
0) to ( k, a k )with the height sequence a ≤ . . . ≤ a k , we say that an increasing lattice path u with theheight sequence b ≤ . . . ≤ b l is below v if the two paths share the same starting point(0 ,
0) and the same ending point ( k, a k ) (hence l = k ) and 0 ≤ b i ≤ a i for i = 1 , . . . , k. Proposition 3.3.
Let v be a given increasing lattice path in the closed first quadrantfrom (0 , to ( k, a k ) with the height sequence a ≤ . . . ≤ a k . Then the number l k ofincreasing lattice paths below v is l k = det ≤ i,j ≤ k (cid:18)(cid:18) a i + 1 j − i + 1 (cid:19)(cid:19) . Connecting Theorem 3.2 to Proposition 3.3, we obtain a combinatorial identity.
Corollary 3.4.
For a sequence λ ≥ · · · ≥ λ k of nonnegative integers, we have det ≤ i,j ≤ k (cid:18) λ i + 1 i − j + 1 (cid:19) = k − X i =1 (cid:18) λ i + k − i + 1 k + 1 − i (cid:19) γ i − k − X i =1 (cid:18) λ i − λ k + k − ik + 1 − i (cid:19) γ i − k − X i =1 ( λ k + 1) (cid:18) λ i − λ k − + k − i − k − i (cid:19) γ i , where γ i is given in (3.1).Proof. Let a i = λ k − i +1 . Then a ≤ · · · ≤ a k . It follows from Proposition 3.3 that thenumber l k of increasing lattice paths below a ≤ · · · ≤ a k is l k = det ≤ i,j ≤ k (cid:18)(cid:18) λ k − i +1 + 1 j − i + 1 (cid:19)(cid:19) . D be the anti-diagonal matrix of size k with 1 for each entry on the anti-diagonaland all other entries 0. If A = ( a ij ) is any matrix of size k , then DAD = ( a k − i +1 , k − j +1 )and det A = det DAD.
Taking A = (cid:0)(cid:0) λ k − i +1 +1 j − i +1 (cid:1)(cid:1) , we find that DAD = (cid:0)(cid:0) λ i +1 i − j +1 (cid:1)(cid:1) . Itfollows that l k = det ≤ i,j ≤ k (cid:18)(cid:18) λ i + 1 i − j + 1 (cid:19)(cid:19) . By symmetry, this number equals the number d k of decreasing lattice paths belowthe lattice path from (0 , λ ) to ( k,
0) with the height sequence λ ≥ · · · ≥ λ k . UsingTheorem 3.2, we obtain the required result.We have the combinatorial identity below for the Catalan number c n = n +1 (cid:0) nn (cid:1) . Toour knowledge, the identity is new. Corollary 3.5. If k ≥ , then c k +1 = k − X i =1 (cid:18) k − i + 1) k + 1 − i (cid:19) γ i − k − X i =1 (cid:18) k − i ) k + 1 − i (cid:19) γ i − k − X i =1 (cid:18) k − i − k − i (cid:19) γ i , where γ = 1 and for ≤ i ≤ k , γ i = − i − X j =1 (cid:18) i − j − i − j (cid:19) γ j . Proof.
Let ( λ ≥ λ ≥ · · · ≥ λ k ) = ( k ≥ ( k − ≥ · · · ≥ µ ≥ µ ≥ . . . ≥ µ k of nonnegative integers such that µ i ≤ λ i for 1 ≤ i ≤ k isthe Catalan number c k +1 . Simplifying the formula for d k in Theorem 3.2, we completethe proof. Corollary 3.6.
Let ( λ ≥ · · · ≥ λ k ≥ be a given partition of some nonnegative integer.Then the number of distinct Young diagrams, with each row having equal or fewer boxesthan the row above, obtained from the Young diagram of λ by removing zero or moreboxes from the rows is d k .Proof. It is easily seen that the number of the desired distinct Young diagrams obtainedfrom the Young diagram of λ by removing zero or more boxes from the rows is equal tothe number of sequences µ ≥ · · · ≥ µ k ≥ µ i ≤ λ i for all i = 1 , , . . . , k . Theresult follows from Lemma 3.1 and Theorem 3.2. I C n Our aim of this section is to apply Theorem 3.2 for enumerating lattice paths tocalculate the dimensions of submodules for IC n . To this end we need some preparationsto describe precisely the structure of the modules involved; our results go a little deeperand wider than just for calculating the dimensions.8 .1 Properties of modules over IC n A vector space V over a field F of characteristic 0 is called an IC n -module if IC n acts on V satisfying, for all f, f , f ∈ IC n , u, v ∈ V , and λ ∈ F , f · ( u + v ) = f · u + f · v, f · ( f · u ) = ( f f ) · u,f · ( λu ) = λ ( f · u ) , · u = u. From now on, V denotes a vector space with a basis B = { v S | S ⊆ n } indexed by allthe subsets of n . Then V = L S ⊆ n F v S as subspaces is an IC n -module with respect tothe following action: for f ∈ IC n and S ⊆ n , f · v S = (cid:26) v S ′ , if S ⊆ D ( f )0 , otherwise,where S ′ = { f ( s ) , . . . , f ( s k ) } if S = { s , . . . , s k } . For 0 ≤ k ≤ n , let V k = span { v S ∈ B | k = | S |} . Then V = L nk =0 V k is a direct sum of IC n -submodules.Every module under consideration is an IC n -module over F , unless otherwise stated.To describe the IC n -module structure of V k and V , we define a partial order on the powerset of n . For any k -subsets S = { s < · · · < s k } and T = { t < · · · < t k } of n , define T ≤ S ⇔ t i ≤ s i for all i ∈ k , and a k -subset is not comparable to any l -subset if k = l .For v ∈ V we use h v i to denote the cyclic submodule of V generated by v . If S is a k -subset of n , then h v S i is a submodule of V k . Indeed, for any f ∈ IC n if S ⊆ D ( f ) then f ( S ) is a k -subset, so f · v S = v f ( S ) ∈ V k ; if S is not a subset of D ( f ) then f · v S = 0 ∈ V k .Some further properties of the module h v S i are described in the next result. Lemma 4.1.
Let
S, T be k -subsets of n . (1) h v S i = L S ′ ⊆ n , S ′ ≤ S F v S ′ as vector spaces. In particular, V k = h v { n − k +1 , ..., n } i . (2) h v T i ⊆ h v S i if and only if T ≤ S . (3) h v S i ∩ h v T i = h v S ∧ T i , where S ∧ T is the greatest lower bound of S and T .Proof. To prove (1) notice that two subsets S ′ ≤ S if and only if S = D ( f ) and S ′ = R ( f )for a unique f ∈ IC n . Let S ′ ≤ S . Then v S ′ = f · v S ∈ h v S i . Hence L S ′ ⊆ n , S ′ ≤ S F v S ′ isincluded in h v S i . Conversely, let x = g · v S = 0 for some g ∈ IC n . We have S ⊆ D ( g ), g ( S ) ≤ S , and hence x = v g ( S ) ∈ L S ′ ⊆ n , S ′ ≤ S F v S ′ . The second part of (i) is now clear.The proof of (2) follows from (1) since { T ′ | T ′ ⊆ n , T ′ ≤ T } ⊆ { S ′ | S ′ ⊆ n , S ′ ≤ S } if and only if T ≤ S .To prove (3) let g · v S = h · v T = 0 for some g, h ∈ IC n . Then g ( S ) = h ( T ). Suppose S = { s < . . . < s k } and T = { t < . . . < t k } . S ∧ T = { min( s , t ) , . . . , min( s k , t k ) } , and g ( s i ) = h ( t i ). We define f ∈ IC n with D ( f ) = S ∧ T and R ( f ) = g ( S ) by f (min( s i , t i )) = g ( s i ) , where 1 ≤ i ≤ k .Then g · v S = f · v S ∧ T ∈ h v S ∧ T i , and hence h v S i ∩ h v T i ⊆ h v S ∧ T i . Conversely, for anygiven 0 = f · v S ∧ T ∈ h v S ∧ T i define g ( s i ) = h ( t i ) = f (min( s i , t i )) for 1 ≤ i ≤ k . Then f · v S ∧ T = g · v S = h · v T ∈ h v S i ∩ h v T i . The proof of (3) is complete.Let v = P S ⊆ n λ S v S , λ S ∈ F be a vector of V . The support of v is defined to besupp( v ) = { S ⊆ n | λ S = 0 } . Definition 4.2.
A vector of the form w = P S ∈ supp( w ) v S ∈ V is called a reduced gener-ator of a submodule W of V if W = h w i and W cannot be generated by any other vectorwhose support contains fewer elements than supp( w ) . We agree that is the reducedgenerator of the zero submodule. The next proposition gives some properties of submodules of V . Proposition 4.3.
Let v = P S ∈ supp( v ) λ S v S ∈ V . (1) If S is in supp( v ) , then v S ∈ h v i . (2) h v i = L T ∈P ( v ) F v T as subspaces, where P ( v ) = S S ∈ supp( v ) { T ⊆ n | T ≤ S } . (3) Every submodule of V is cyclic and contains a unique reduced generator.Proof. To prove (1) let min (cid:8) | S | (cid:12)(cid:12) S ∈ supp( v ) (cid:9) = r . Then there exists an r -subset T = { t < · · · < t r } ⊆ n such that T ∈ supp( v ); if r = 0, then T = ∅ . Let f ∈ IC n suchthat D ( f ) = R ( f ) = T . By the choice of r , for every S ∈ supp( v ) with S = T , there isat least one s ∈ S such that s / ∈ T , so f · v S = 0. Hence f · v = f · X S ∈ supp( v ) λ S v S = X S ∈ supp( v ) λ S ( f · v S ) = λ T v T . Thus v T ∈ h v i since λ T = 0. It is easily seen that X S ∈ supp( v ) | S | >r λ S v S = v − X S ∈ supp( v ) | S | = r λ S v S ∈ h v i . Applying the above procedure to P S ∈ supp( v ) , | S | >r λ S v S and iteratively using this proce-dure, if needed, we get v S ⊆ h v i for all S ∈ supp( v ). The proof of (1) is complete.From (1) and Lemma 4.1 (1), we have h v i = X S ∈ supp( v ) λ S h v S i = X S ∈ supp( v ) span { v T ∈ B | T ≤ S } = M S ∈P ( v ) F v S , as subspaces . W = { } . Let W be a nonzero submodule of V .We claim that W has a basis { v S ∈ B | S ∈ P} for some subset P of the power set of n .Indeed, suppose B is a basis of W and write every element of B as a linear combinationof basis vectors in B = { v S | S ⊆ n } . Let P be the set of all the different subsets S where S runs through the support of every element of B . By (1) the set { v S ∈ B | S ∈ P} is asubset of W , and hence a basis of W since it is linearly independent and spans W . Let w = P S ∈P v S . By (1) again, W is generated by w , and hence W is cyclic.We now show how to deduce a reduced generator of W from w . Indeed, if w containstwo vectors v S , v T with T ≤ S and T = S in supp( w ), then we can remove the term v T from w , and by Lemma 4.1 (i) the sum of the remaining terms is still a generator.Repeat this process until we obtain the setRed( w ) = { S | S is maximal in supp( w ) } , and then we define the corresponding generator w red of W by w red = X S ∈ Red( w ) v S . We claim that w red is a reduced generator of W . Let v = P S ∈ supp( v ) λ S v S be anothergenerator of W . From Definition 4.2 it suffices to show that | supp( v ) | ≥ | Red( w ) | . From(2) we find W = L T ∈P ( v ) F v T = L T ∈P ( w ) F v T where P ( v ) and P ( w ) are as in (2), andhence P ( v ) = P ( w ) . DefineRed( v ) = { S | S is maximal in supp( v ) } . (4.1)Thus, Red( v ) = { S | S is maximal in P ( v ) } and Red( w ) = { S | S is maximal in P ( w ) } .So, Red( v ) = Red( w ) and | supp( v ) | ≥ | Red( v ) | = | Red( w ) | , showing that w red is reduced.Suppose that v = P S ∈ supp( v ) v S is another reduced generator of W . By the definitionof reduced generators we know | supp( v ) | = | Red( w ) | . Hence | supp( v ) | = | Red( v ) | sinceRed( v ) = Red( w ). It follows that supp( v ) = Red( v ). Let v red = P S ∈ Red( v ) v S . Then v = v red = w red . Therefore w red is the unique reduced generator of W . Definition 4.4.
The set
Red( v ) in (4.1) is called the reduced support of v , and theelement v red = P S ∈ Red( v ) v S is termed the reduced form of v . The reduced support of is empty, and the reduced form of is itself. For example, if n = 7 and v = v ∅ − v { } + v { } + 5 v { , } + 3 v { , } − v { , } + v { , , } ,then Red( v ) = {∅ , { } , { , } , { , } , { , , }} is the reduced support of v , and itsreduced form is v red = v ∅ + v { } + v { , } + v { , } + v { , , } .It is sometimes convenient to call the reduced support of v the reduced support of themodule h v i . A direct calculation yields that the reduced generator of V k is v { n − k +1 , ..., n } for 1 ≤ k ≤ n , and the reduced support of V k is the set { n − k + 1 , . . . , n } . The module V has the element v ∅ as its reduced generator, and its reduced support is the set {∅} .The next result is a consequence of Lemma 4.1 (i) and Proposition 4.3 (3).11 orollary 4.5. If v, w ∈ V , then h v i = h w i if and only if they have the same reducedsupport Red( v ) = Red( w ) if and only if they have the same reduced generator v red = w red . V We now describe the dimension of h v S i for any S ⊆ n . Theorem 4.6. If S = { s < · · · < s k } is a k -subset of n , let d S be the dimension of themodule h v S i . If k = 1 then d S = s , and for k ≥ , d S = k − X i =1 (cid:18) s k − i +1 k + 1 − i (cid:19) γ i − k − X i =1 (cid:18) s k − i +1 − s k + 1 − i (cid:19) γ i − k − X i =1 s (cid:18) s k − i +1 − s k − i (cid:19) γ i , (4.2) where γ = 1 and for ≤ j ≤ k − , γ j = − j − X i =1 (cid:18) s k +1 − i − s k +2 − j j − i (cid:19) γ i . Proof.
By Lemma 4.1 (i) we know that d S is equal to the number of k -subsets T of n such that T ≤ S . Let λ i = s k − i +1 − ( k − i + 1) for 1 ≤ i ≤ k . (4.3)Then λ i ≥ λ i +1 since s k − i +1 > s k − i . Because the smallest k -subset is { , . . . , k } , we have λ ≥ · · · ≥ λ k ≥ , (4.4)and the number of k -subsets T of n with T ≤ S is equal to the number d k of all thesequences µ ≥ · · · ≥ µ k ≥ µ i ≤ λ i for i = 1 , . . . , k . Thus d S = d k . From Lemma 3.1 and Theorem 3.2 we have d = λ + 1, and for k ≥ d k = k − X i =1 (cid:20)(cid:18) λ i + k − i + 1 k + 1 − i (cid:19) − (cid:18) λ i − λ k + k − ik + 1 − i (cid:19)(cid:21) γ i − k − X i =1 ( λ k + 1) (cid:18) λ i − λ k − + k − i − k − i (cid:19) γ i , where γ = 1 and γ j = − j − X i =1 (cid:18) λ i − λ j − + j − i − j − i (cid:19) γ i for j ≥ . Using (4.3), we conclude that (4.2) holds.
Corollary 4.7. If S = { , , . . . , k } ⊆ n , then the dimension of the submodule h v S i isthe Catalan number c k +1 .Proof. The sequence λ ≥ · · · ≥ λ k ≥ S is now k ≥ k − ≥ · · · ≥ µ ≥ µ ≥ . . . ≥ µ k ofnonnegative integers such that µ i ≤ λ i for 1 ≤ i ≤ k is the Catalan number c k +1 . Thedesired result follows from Theorem 4.6. 12 orollary 4.8. If the k -subset S = { m + 1 , . . . , m + k } ⊆ n , the dimension of thesubmodule h v S i is (cid:0) m + kk (cid:1) .Proof. The sequence in (4.4) corresponding to S is the k -subset { m, . . . , m } . A directcalculation of d S for k ≥ d k = (cid:18) m + kk (cid:19) , which is the desired result.We now compute the dimension d k, m of the IC n -module h v S k, m i , where k ≥ m and S k, m = { , , . . . , m, m + 1 , m + 2 , . . . , m + ( k − m ) } is a subset of n , which consists of both subsets in Corollaries 4.7 and 4.8. Corollary 4.9.
For m ≥ the dimension of the module h v S k, m i is d k, m = (cid:18) m + kk (cid:19) − (cid:18) m + k − k (cid:19) − (cid:18) m + k − k − (cid:19) + k − X i = k − m +3 (cid:18) k − i + 1) k + 1 − i (cid:19) γ i − k − X i = k − m +3 (cid:18) k − i ) k + 1 − i (cid:19) γ i − k − X i = k − m +3 (cid:18) k − i − k − i (cid:19) γ i where γ = 1 , γ = γ = · · · = γ k − m +2 = 0 , γ k − m +3 = − , and for i ≥ k − m + 4 γ i = − (cid:18) m − k + 2 i − i − (cid:19) − i − X j = k − m +3 (cid:18) i − j − i − j (cid:19) γ j . Proof.
The sequence (4.4) associated to S k, m is { m, m, . . . , m, m − , m − , . . . , } of length k . The desired formula follows from Theorem 4.6, and we omit the tediouscalculation. Remark 4.10.
Notice that d k, k is exactly the Catalan number c k +1 by Corollary (4.7). The intention below is to give another description of the dimensions of h v S i usingProposition 3.3. Theorem 4.11. If S = { s < · · · < s k } is a k -subset of n and d S is the dimension ofthe module h v S i , then d S = X T = { t Let v ∈ V and Red( v ) = { S , . . . , S m } . For any J ⊆ Red( v ) denote by S J the greatest lower bound of { S j | j ∈ J } . Then the dimension of h v i is given by dim h v i = X ∅ 6 = J ⊆ m ( − | J |− dim h v S J i . Proof.