The Gaussian coefficient revisited

Transcription

1 The Gaussian coefficient revisited Richard EHRENBORG and Margaret A. READDY Abstract We give new -(1+)-analogue of the Gaussian coefficient, also now as the -binomial which, lie the original -binomial [ ] n, is symmetric in and n. We show this -(1 + )-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our -(1 + )-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birhoff transform of any finite poset. We end with a discussion of higher analogues of the -binomial Mathematics Subject Classification. Primary 06A07; Secondary 05A05, 05A10, 05A30. Key words and phrases. -analogue, Birhoff transform, distributive lattice, poset decomposition. 1 Introduction Inspired by wor of Fu, Reiner, Stanton and Thiem [2], Cai and Readdy [1] ased the following uestion. Given a combinatorial -analogue X() w X a(w), where X is a set of objects and a( ) is a statistic defined on the elements of X, when can one find a smaller set Y and two statistics s and t such that X() w Y s(w) (1 + ) t(w). Such an interpretation is called an -(1 + )-analogue. Examples of -(1 + )-analogues have been determined for the -binomial by Fu, Reiner, Stanton and Thiem [2], and for the -Stirling numbers of the first and second inds by Cai and Readdy [1], who also gave poset and homotopy interpretations of their -(1 + )-analogues. Corresponding author: Department of Mathematics, University of Kentucy, Lexington, KY , USA, richard.ehrenborg@uy.edu. Department of Mathematics, University of Kentucy, Lexington, KY , USA, margaret.readdy@uy.edu. 1

2 In 1916 MacMahon [3, 4, 5] observed that the Gaussian coefficient, also nown as the -binomial coefficient, is given by inv(w). w Ω n, Here Ω n, = S(0 n, 1 ) denotes all permutations of the multiset {0 n, 1 }, that is, all words w = w 1 w n of length n with n zeroes and ones, and inv( ) denotes the inversion statistic defined by inv(w 1 w 2 w n ) = {(i, j) : 1 i < j n, w i > w j }. Fu et al. defined a subset Ω n, Ω n, and two statistics a and b such that a(w) (1 + ) b(w). w Ω n, In this paper we will return to the original study by Fu et al. of the Gaussian coefficient. We discover a more compact -(1 + )-analogue which, lie the original Gaussian coefficients, is also symmetric in the variables and n ; see Corollary 2.6 and Theorem 3.5. This symmetry was missing in Fu et al. s original -(1 + )-analogue. We give a Boolean algebra decomposition of the related poset Ω n,. Since this poset is a distributive lattice, in the last section we extend these ideas to poset decompositions of any distributive lattice and other analogues. 2 A poset interpretation In this section we consider the poset structure on 0-1-words in Ω n,. For further poset terminology and bacground, we refer the reader to [6]. We begin by maing the set of elements Ω n, into a graded poset by defining the cover relation to be u 01 v u 10 v, where denotes concatenation of words. The word 0 n 1 is the minimal element and the word 1 0 n is the maximal element in the poset Ω n,. Furthermore, this poset is graded by the inversion statistic. This poset is simply the interval [ 0, x] of Young s lattice, where the minimal element 0 is the empty Ferrers diagram and x is the Ferrers diagram consisting of n columns and rows. An alternative description of the poset Ω n, is that it is isomorphic to the Birhoff transform of the Cartesian product of two chains. Let C m denote the m-element chain. The poset Ω n, is isomorphic to the distributive lattice of all lower order ideals of the product C n C, usually denoted by J(C n C ). Definition 2.1. Let Ω n, Ω n, consist of all 0,1-words v = v 1 v 2 v n in Ω n, such that v 1 v 2, v 3 v 4,..., v 2 n/2 1 v 2 n/2. Observe that when n is odd there is no condition on the last entry w n. Define two maps φ and ψ on Ω n, by sending the word w = w 1 w 2 w n to φ(w) = min(w 1, w 2 ), max(w 1, w 2 ), min(w 3, w 4 ), max(w 3, w 4 ),..., ψ(w) = max(w 1, w 2 ), min(w 1, w 2 ), max(w 3, w 4 ), min(w 3, w 4 ),.... 2

3 The map φ sorts the entries in positions 1 and 2, 3 and 4, and so on. If n is odd, the entry w n remains in the same position. Similarly, the map ψ sorts in reverse order in each pair of positions. Note that the map φ maps Ω n, surjectively onto the set Ω n,. We have the following Boolean algebra decomposition of the poset Ω n,. Theorem 2.2. The distributive lattice Ω n, has the Boolean algebra decomposition Ω n, = v Ω n, [v, ψ(v)]. Proof. Observe that the maps φ and ψ satisfy the ineualities φ(w) w ψ(w). Furthermore, the fiber of the map φ : Ω n, Ω n, is isomorphic to a Boolean algebra, that is, φ 1 (v) = [v, ψ(v)]. For v Ω n, define the statistic asc odd (v) = {i : v i < v i+1, i odd}, that is, asc odd ( ) enumerates the number of ascents in odd positions. Corollary 2.3. The -binomial is given by v Ω n, inv(v) (1 + ) asc odd(v). (2.1) Proof. It is enough to observe that the sum of the inversion statistic over the elements in the fiber φ 1 (v) = [v, ψ(v)] for v Ω n, is given by inv(v) (1 + ) asc odd(v). A geometric way to understand this -(1 + )-interpretation is to consider lattice paths from the origin (0, 0) to (n, ) which only use east steps (1, 0) and north steps (0, 1). Color the suares of this (n ) board as a chessboard, where the suare incident to the origin is colored white. The map φ in the proof of Theorem 2.2 corresponds to taing a lattice path where every time there is a north step followed by an east step that turns around a white suare, we exchange these two steps. The statistic asc odd enumerates the number of times an east step is followed by a north step when this pair of steps borders a white suare. Let er(n, ) denote the cardinality of the set Ω n,. Then we have Proposition 2.4. The cardinalities er(n, ) satisfy the recursion er(n, ) = er(n 2, 2) + er(n 2, 1) + er(n 2, ) for n, 2 with er(n, n) = 1 and er(n, ) = 0 whenever > n, < 0 or n < 0. Proof. A word in Ω n, begins with either 00, 01 or 11, yielding the three cases of the recursion. 3

4 Directly we obtain the generating polynomial. Theorem 2.5. The generating polynomial for er(n, ) is given by n er(n, ) x = (1 + x + x 2 ) n/2 (1 + x) n 2 n/2. =0 We end with a statement concerning the symmetry of the -(1 + )-binomial. Corollary 2.6. The set of defining elements for the -(1 + )-binomial satisfy the following symmetric relation: Ω n, = Ω n,n. Proof. This follows from the fact that the generating polynomial for er(n, ) is a product of palindromic polynomials, and thus is itself is a palindromic polynomial. 3 Analysis of the Fu Reiner Stanton Thiem interpretation A wea partition is a finite non-decreasing seuence of non-negative integers. A wea partition λ = (λ 1,..., λ n ) with n parts and each part at most where λ 1 λ n corresponds to a Ferrers diagram lying inside an (n ) rectangle with column i having height λ i. These wea partitions are in direct correspondence with the set Ω n,. Fu, Reiner, Stanton and Thiem used a pairing algorithm to determine a subset Ω n, Ω n, of 0-1-seuences to define their -(1 + )-analogue of the -binomial; see [2, Proposition 6.1]. This translates into the following statement. The set Ω n, is in bijection with wea partitions into n parts with each part at most such that (a) if is even, each odd part has even multiplicity, (b) if is odd, each even part (including 0) has even multiplicity. Definition 3.1. Let frst(n, ) be the cardinality of the set Ω n,. Lemma 3.2. The uantity frst(n, ) counts the number of wea partitions into n parts where each part is at most and each odd part has even multiplicity. Proof. When is even there is nothing to prove. When is odd, by considering the complement of wea partitions with respect to the rectangle of size (n ), we obtain a bijective proof. The same complement proof also shows the case when is even holds. Theorem 3.3. The frst-coefficients satisfy the recursion frst(n, ) = frst(n 1, 1) + frst(n 1, ) frst(n, ) = frst(n 2, 2) + frst(n 2, 1) + frst(n 2, ) for even, for odd, where 1 n 1. 4

5 Proof. We use the characterization in Lemma 3.2. When is even there are two cases. If the last part is, remove it to obtain a wea partition counted by frst(n 1, ). If the last part is less than, then the wea partition is counted by frst(n 1, 1). When is odd there are three cases. If the last two parts are eual to, then removing these two parts yields a wea partition counted by frst(n 2, ). Note that we cannot have the last part eual to and the next to last part less than since is odd. If the last part is eual to 1, we can remove it to obtain a wea partition counted by frst(n 2, 1). Finally, if the last part is less than or eual to 2, the wea partition is counted by frst(n 2, 2). Lemma 3.4. The ineuality frst(n, ) frst(n + 1, + 1) holds. Proof. The wea partitions which lie inside the rectangle (n ) and satisfy the conditions of Lemma 3.2 are included among the wea partitions which lie inside the larger rectangle (n ) ( + 1) and satisfy the same conditions. Theorem 3.5. For all 0 n the ineuality Ω n, = er(n, ) frst(n, ) = Ω n, holds. Proof. We proceed by induction on n. The induction base is n 3. Furthermore, the ineuality holds when is 0, 1, n 1 and n. When is odd we have that er(n, ) = er(n 2, 2) + er(n 2, 1) + er(n 2, ) frst(n 2, 2) + frst(n 2, 1) + frst(n 2, ) = frst(n, ). Similarly, when is even we have er(n, ) = er(n 2, 2) + er(n 2, 1) + er(n 2, ) frst(n 2, 2) + frst(n 2, 1) + frst(n 2, ) frst(n 1, 1) + frst(n 2, 1) + frst(n 2, ) = frst(n 1, 1) + frst(n 1, ) = frst(n, ), where the second ineuality follows from Lemma 3.4. hypothesis. These two cases complete the induction See Table 1 to compare the values of frst(n, ) and er(n, ) for n Concluding remars Is it possible to find a -(1 + )-analogue of the Gaussian coefficient which has the smallest possible index set? We believe that our analogue is the smallest, but cannot offer a proof of a minimality. Perhaps a more tractable uestion is to prove that the Boolean algebra decomposition of Ω n, is minimal. 5

6 Table 1: The frst- and er-triangles for n 10. We can extend these ideas involving of a Boolean algebra decomposition to any distributive lattice. Let P be a finite poset and let A be an antichain of P such that there is no cover relation in A, that is, there is no pair of elements u, v A such that u v. We obtain a Boolean algebra decomposition of the Birhoff transform J(P ) by defining J (P ) = {I J(P ) : The two maps φ and ψ are now defined as φ(i) = I {a A : the ideal I has no maximal elements in the antichain A}. the element a is maximal in I}, ψ(i) = I {a A : I {a} J(P )}. Then we have the following decomposition theorem Theorem 4.1. For P any finite poset the distributive lattice J(P ) has the Boolean algebra decomposition J(P ) = [I, ψ(i)]. I J (P ) Yet again, how can we select the antichain A such that the above decomposition A has the fewest possible terms? Furthermore, would this give the smallest Boolean algebra decomposition? Another way to extend the ideas of Theorem 2.2 is as follows. Define Ω r n, to be the set of all words v Ω n, satisfying the ineualities v 1 v 2 v r, v r+1 v r+2 v 2r,..., v r n/r r+1 v r n/r r+2 v r n/r. For 1 i r/2 define the statistics b i (v) for v Ω r n, to be b i (v) = {j [ n/r ] : v rj r+1 + v rj r v rj {i, r i}}. Theorem 4.2. The distributive lattice Ω n, has the decomposition Ω n, = Ω b 1(v) r,1 Ω b 2(v) r,2 Ω b r/2 (v) r, r/2. v Ω r n, 6

7 Corollary 4.3. The -binomial is given by inv(v) v Ω r n, [ ] r b1 (v) 1 [ ] r b2 (v) [ ] r b r/2 (v). 2 r/2 The least complicated case is when r = 3, where only one term appears in the above poset product. This term is Ω 3,1 which is the three element chain C 3. The associated Gaussian coefficient is Thus Corollary 4.3 could be called a -( )-analogue. As an example we have [ ] 6 = 1 + ( ) ( ) On a poset level this a decomposition of J(C 3 C 3 ) into two one-element posets of ran 0 and ran 9, and two copies of C 3 C 3, where one has its minimal element of ran 1 and the other of ran 4. Acnowledgements This wor was partially supported by a grant from the Simons Foundation (# to Margaret Readdy). References [1] Y. Cai and M. Readdy, -Stirling numbers: A new view, arxiv: [math.co], 30 pp. [2] S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative -binomial, Electron J. Combin. 19 (2012), 36 pages. [3] P. A. MacMahon, The Indices of Permutations and the Derivation Therefrom of Functions of a Single Variable Associated with the Permutations of any Assemblage of Objects, Amer. J. Math., 35 (1913), , in Collected papers. Vol. I. Combinatorics. Mathematicians of Our Time. Edited and with a preface by George E. Andrews. With an introduction by Gian-Carlo Rota, MIT Press, Cambridge, Mass.-London, 1978, pp [4] P. A. MacMahon, Two applications of general theorems in combinatory analysis, Proc. London Math. Soc., 15 (1916), , in Collected papers. Vol. I. Combinatorics. Mathematicians of Our Time. Edited and with a preface by George E. Andrews. With an introduction by Gian-Carlo Rota, MIT Press, Cambridge, Mass.-London, 1978, pp [5] P. A. MacMahon, Combinatory Analysis, Chelsea Publishing Co., New Yor, [6] R. P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Cambridge University Press, Cambridge,