The Gaussian coefficient revisited
|
|
- Ronald Williams
- 5 years ago
- Views:
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,
A multiplicative deformation of the Möbius function for the poset of partitions of a multiset
Contemporary Mathematics A multiplicative deformation of the Möbius function for the poset of partitions of a multiset Patricia Hersh and Robert Kleinberg Abstract. The Möbius function of a partially ordered
More informationMaximizing the descent statistic
Maximizing the descent statistic Richard EHRENBORG and Swapneel MAHAJAN Abstract For a subset S, let the descent statistic β(s) be the number of permutations that have descent set S. We study inequalities
More informationMULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A06 MULTI-ORDERED POSETS Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States lbishop@oxy.edu
More informationThe symmetric group action on rank-selected posets of injective words
The symmetric group action on rank-selected posets of injective words Christos A. Athanasiadis Department of Mathematics University of Athens Athens 15784, Hellas (Greece) caath@math.uoa.gr October 28,
More informationarxiv:math/ v1 [math.co] 10 Nov 1998
A self-dual poset on objects counted by the Catalan numbers arxiv:math/9811067v1 [math.co] 10 Nov 1998 Miklós Bóna School of Mathematics Institute for Advanced Study Princeton, NJ 08540 April 11, 2017
More informationCounting Matrices Over a Finite Field With All Eigenvalues in the Field
Counting Matrices Over a Finite Field With All Eigenvalues in the Field Lisa Kaylor David Offner Department of Mathematics and Computer Science Westminster College, Pennsylvania, USA kaylorlm@wclive.westminster.edu
More informationPartitions, rooks, and symmetric functions in noncommuting variables
Partitions, rooks, and symmetric functions in noncommuting variables Mahir Bilen Can Department of Mathematics, Tulane University New Orleans, LA 70118, USA, mcan@tulane.edu and Bruce E. Sagan Department
More informationUNIMODALITY OF PARTITIONS WITH DISTINCT PARTS INSIDE FERRERS SHAPES
UNIMODALITY OF PARTITIONS WITH DISTINCT PARTS INSIDE FERRERS SHAPES RICHARD P. STANLEY AND FABRIZIO ZANELLO Abstract. We investigate the rank-generating function F λ of the poset of partitions contained
More informationTHE LARGEST INTERSECTION LATTICE OF A CHRISTOS A. ATHANASIADIS. Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin.
THE LARGEST INTERSECTION LATTICE OF A DISCRIMINANTAL ARRANGEMENT CHRISTOS A. ATHANASIADIS Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin. 6 (1997), 229{246] about the \largest" intersection
More informationHomology of Newtonian Coalgebras
Homology of Newtonian Coalgebras Richard EHRENBORG and Margaret READDY Abstract Given a Newtonian coalgebra we associate to it a chain complex. The homology groups of this Newtonian chain complex are computed
More informationMACMAHON S PARTITION ANALYSIS IX: k-gon PARTITIONS
MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Dedicated to George Szeeres on the occasion of his 90th birthday Abstract. MacMahon devoted a significant
More informationare the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1).
Lecture A jacques@ucsd.edu Notation: N, R, Z, F, C naturals, reals, integers, a field, complex numbers. p(n), S n,, b(n), s n, partition numbers, Stirling of the second ind, Bell numbers, Stirling of the
More informationRook Polynomials In Higher Dimensions
Grand Valley State University ScholarWors@GVSU Student Summer Scholars Undergraduate Research and Creative Practice 2009 Roo Polynomials In Higher Dimensions Nicholas Krzywonos Grand Valley State University
More informationODD PARTITIONS IN YOUNG S LATTICE
Séminaire Lotharingien de Combinatoire 75 (2016), Article B75g ODD PARTITIONS IN YOUNG S LATTICE ARVIND AYYER, AMRITANSHU PRASAD, AND STEVEN SPALLONE Abstract. We show that the subgraph induced in Young
More informationA Bijection between Maximal Chains in Fibonacci Posets
journal of combinatorial theory, Series A 78, 268279 (1997) article no. TA972764 A Bijection between Maximal Chains in Fibonacci Posets Darla Kremer Murray State University, Murray, Kentucky 42071 and
More informationPartitions, permutations and posets Péter Csikvári
Partitions, permutations and posets Péter Csivári In this note I only collect those things which are not discussed in R Stanley s Algebraic Combinatorics boo Partitions For the definition of (number) partition,
More informationParking cars after a trailer
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 70(3) (2018), Pages 402 406 Parking cars after a trailer Richard Ehrenborg Alex Happ Department of Mathematics University of Kentucky Lexington, KY 40506 U.S.A.
More informationA Combinatorial Interpretation of the Numbers 6 (2n)! /n! (n + 2)!
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3 A Combinatorial Interpretation of the Numbers 6 (2n)! /n! (n + 2)! Ira M. Gessel 1 and Guoce Xin Department of Mathematics Brandeis
More informationDIFFERENTIAL POSETS SIMON RUBINSTEIN-SALZEDO
DIFFERENTIAL POSETS SIMON RUBINSTEIN-SALZEDO Abstract. In this paper, we give a sampling of the theory of differential posets, including various topics that excited me. Most of the material is taken from
More informationq xk y n k. , provided k < n. (This does not hold for k n.) Give a combinatorial proof of this recurrence by means of a bijective transformation.
Math 880 Alternative Challenge Problems Fall 2016 A1. Given, n 1, show that: m1 m 2 m = ( ) n+ 1 2 1, where the sum ranges over all positive integer solutions (m 1,..., m ) of m 1 + + m = n. Give both
More information, when k is fixed. We give a number of results in. k q. this direction, some of which involve Eulerian polynomials and their generalizations.
SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS RICHARD P STANLEY AND FABRIZIO ZANELLO Abstract We loo at the asymptotic behavior of the coefficients of the -binomial coefficients or Gaussian polynomials,
More informationSUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS
SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P, let Co(P) denote the lattice of all order-convex
More informationA quasisymmetric function generalization of the chromatic symmetric function
A quasisymmetric function generalization of the chromatic symmetric function Brandon Humpert University of Kansas Lawrence, KS bhumpert@math.ku.edu Submitted: May 5, 2010; Accepted: Feb 3, 2011; Published:
More informationYi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian , China (Submitted June 2002)
SELF-INVERSE SEQUENCES RELATED TO A BINOMIAL INVERSE PAIR Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China (Submitted June 2002) 1 INTRODUCTION Pairs of
More informationA Characterization of (3+1)-Free Posets
Journal of Combinatorial Theory, Series A 93, 231241 (2001) doi:10.1006jcta.2000.3075, available online at http:www.idealibrary.com on A Characterization of (3+1)-Free Posets Mark Skandera Department of
More informationarxiv: v2 [math.co] 3 Jan 2019
IS THE SYMMETRIC GROUP SPERNER? arxiv:90.0097v2 [math.co] 3 Jan 209 LARRY H. HARPER AND GENE B. KIM Abstract. An antichain A in a poset P is a subset of P in which no two elements are comparable. Sperner
More informationShellability of Interval Orders
Shellability of Interval Orders Louis J. Billera and Amy N. Myers September 15, 2006 Abstract An finite interval order is a partially ordered set whose elements are in correspondence with a finite set
More information4-Shadows in q-series and the Kimberling Index
4-Shadows in q-series and the Kimberling Index By George E. Andrews May 5, 206 Abstract An elementary method in q-series, the method of 4-shadows, is introduced and applied to several poblems in q-series
More informationTwo-boundary lattice paths and parking functions
Two-boundary lattice paths and parking functions Joseph PS Kung 1, Xinyu Sun 2, and Catherine Yan 3,4 1 Department of Mathematics, University of North Texas, Denton, TX 76203 2,3 Department of Mathematics
More informationThe Descent Set and Connectivity Set of a Permutation
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.8 The Descent Set and Connectivity Set of a Permutation Richard P. Stanley 1 Department of Mathematics Massachusetts Institute
More informationWhat you learned in Math 28. Rosa C. Orellana
What you learned in Math 28 Rosa C. Orellana Chapter 1 - Basic Counting Techniques Sum Principle If we have a partition of a finite set S, then the size of S is the sum of the sizes of the blocks of the
More informationA NOTE ON SOME MAHONIAN STATISTICS
Séminaire Lotharingien de Combinatoire 53 (2005), Article B53a A NOTE ON SOME MAHONIAN STATISTICS BOB CLARKE Abstract. We construct a class of mahonian statistics on words, related to the classical statistics
More informationEuler characteristic of the truncated order complex of generalized noncrossing partitions
Euler characteristic of the truncated order complex of generalized noncrossing partitions D. Armstrong and C. Krattenthaler Department of Mathematics, University of Miami, Coral Gables, Florida 33146,
More informationTitle: Equidistribution of negative statistics and quotients of Coxeter groups of type B and D
Title: Euidistribution of negative statistics and uotients of Coxeter groups of type B and D Author: Riccardo Biagioli Address: Université de Lyon, Université Lyon 1 Institut Camille Jordan - UMR 5208
More informationSymmetric chain decompositions of partially ordered sets
Symmetric chain decompositions of partially ordered sets A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Ondrej Zjevik IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
More informationTHE LECTURE HALL PARALLELEPIPED
THE LECTURE HALL PARALLELEPIPED FU LIU AND RICHARD P. STANLEY Abstract. The s-lecture hall polytopes P s are a class of integer polytopes defined by Savage and Schuster which are closely related to the
More informationThe van der Waerden complex
The van der Waerden complex Richard EHRENBORG, Likith GOVINDAIAH, Peter S. PARK and Margaret READDY Abstract We introduce the van der Waerden complex vdw(n, k) defined as the simplicial complex whose facets
More informationINTERVAL PARTITIONS AND STANLEY DEPTH
INTERVAL PARTITIONS AND STANLEY DEPTH CSABA BIRÓ, DAVID M. HOWARD, MITCHEL T. KELLER, WILLIAM. T. TROTTER, AND STEPHEN J. YOUNG Abstract. In this paper, we answer a question posed by Herzog, Vladoiu, and
More informationSergey Fomin* and. Minneapolis, MN We consider the partial order on partitions of integers dened by removal of
Rim Hook Lattices Sergey Fomin* Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139 Theory of Algorithms Laboratory St. Petersburg Institute of Informatics Russian Academy
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 11
18.312: Algebraic Combinatorics Lionel Levine Lecture date: March 15, 2011 Lecture 11 Notes by: Ben Bond Today: Mobius Algebras, µ( n ). Test: The average was 17. If you got < 15, you have the option to
More informationCounting Peaks and Valleys in a Partition of a Set
1 47 6 11 Journal of Integer Sequences Vol. 1 010 Article 10.6.8 Counting Peas and Valleys in a Partition of a Set Toufi Mansour Department of Mathematics University of Haifa 1905 Haifa Israel toufi@math.haifa.ac.il
More informationarxiv: v1 [math.co] 20 Dec 2016
F-POLYNOMIAL FORMULA FROM CONTINUED FRACTIONS MICHELLE RABIDEAU arxiv:1612.06845v1 [math.co] 20 Dec 2016 Abstract. For cluster algebras from surfaces, there is a known formula for cluster variables and
More informationIsomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
More informationSet partition statistics and q-fibonacci numbers
Set partition statistics and q-fibonacci numbers Adam M. Goyt Department of Mathematics Michigan State University East Lansing, Michigan 48824-1027 goytadam@msu.edu www.math.msu.edu/ goytadam Bruce E.
More informationPARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS
PARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS GEORGE E. ANDREWS, MATTHIAS BECK, AND NEVILLE ROBBINS Abstract. We study the number p(n, t) of partitions of n with difference t between
More informationA Major Index for Matchings and Set Partitions
A Major Index for Matchings and Set Partitions William Y.C. Chen,5, Ira Gessel, Catherine H. Yan,6 and Arthur L.B. Yang,5,, Center for Combinatorics, LPMC Nankai University, Tianjin 0007, P. R. China Department
More informationMATH 802: ENUMERATIVE COMBINATORICS ASSIGNMENT 2
MATH 80: ENUMERATIVE COMBINATORICS ASSIGNMENT KANNAPPAN SAMPATH Facts Recall that, the Stirling number S(, n of the second ind is defined as the number of partitions of a [] into n non-empty blocs. We
More informationEnumerating multiplex juggling patterns
Enumerating multiplex juggling patterns Steve Butler Jeongyoon Choi Kimyung Kim Kyuhyeok Seo Abstract Mathematics has been used in the exploration and enumeration of juggling patterns. In the case when
More informationAcyclic Digraphs arising from Complete Intersections
Acyclic Digraphs arising from Complete Intersections Walter D. Morris, Jr. George Mason University wmorris@gmu.edu July 8, 2016 Abstract We call a directed acyclic graph a CI-digraph if a certain affine
More informationCOMPOSITIONS, PARTITIONS, AND FIBONACCI NUMBERS
COMPOSITIONS PARTITIONS AND FIBONACCI NUMBERS ANDREW V. SILLS Abstract. A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions
More informationCIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December Points Possible
Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December 2016 Question Points Possible Points Received 1 12 2 14 3 14 4 12 5 16 6 16 7 16 Total 100 Instructions: 1. This exam
More informationMA 524 Final Fall 2015 Solutions
MA 54 Final Fall 05 Solutions Name: Question Points Score 0 0 3 5 4 0 5 5 6 5 7 0 8 5 Total: 60 MA 54 Solutions Final, Page of 8. Let L be a finite lattice. (a) (5 points) Show that p ( (p r)) (p ) (p
More informationNegative q-stirling numbers
Discrete Mathematics and Theoretical Computer Science DMTCS vol. (subm.), by the authors, 1 1 Negative q-stirling numbers Yue Cai 1 and Margaret Readdy 1 1 University of Kentucky, Department of Mathematics,
More informationPolynomials with palindromic and unimodal coefficients
Polynomials with palindromic and unimodal coefficients arxiv:60.05629v [math.co] 2 Jan 206 Hua Sun, Yi Wang, Hai-Xia Zhang School of Mathematical Sciences, Dalian University of Technology, Dalian 6024,
More informationarxiv:math/ v1 [math.co] 27 Nov 2006
arxiv:math/0611822v1 [math.co] 27 Nov 2006 AN EXTENSION OF THE FOATA MAP TO STANDARD YOUNG TABLEAUX J. HAGLUND,1 AND L. STEVENS Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395,
More informationSequences that satisfy a(n a(n)) = 0
Sequences that satisfy a(n a(n)) = 0 Nate Kube Frank Ruskey October 13, 2005 Abstract We explore the properties of some sequences for which a(n a(n)) = 0. Under the natural restriction that a(n) < n the
More informationA NATURAL EXTENSION OF THE YOUNG PARTITIONS LATTICE
A NATURAL EXTENSION OF THE YOUNG PARTITIONS LATTICE C. BISI, G. CHIASELOTTI, G. MARINO, P.A. OLIVERIO Abstract. Recently Andrews introduced the concept of signed partition: a signed partition is a finite
More information1. Introduction
Séminaire Lotharingien de Combinatoire 49 (2002), Article B49a AVOIDING 2-LETTER SIGNED PATTERNS T. MANSOUR A AND J. WEST B A LaBRI (UMR 5800), Université Bordeaux, 35 cours de la Libération, 33405 Talence
More informationON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY
ON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY YUFEI ZHAO Abstract. In this paper we discuss the Bruhat order of the symmetric group. We give two criteria for comparing elements in this
More informationOn the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers
Séminaire Lotharingien de Combinatoire XX (2019) Article #YY, 12 pp. Proceedings of the 31 st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana) On the Homogenized Linial Arrangement:
More informationSome Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh , India
Some Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh - 160010, India Abstract. Anand and Agarwal, (Proc. Indian Acad. Sci.
More informationRHOMBUS TILINGS OF A HEXAGON WITH TWO TRIANGLES MISSING ON THE SYMMETRY AXIS
RHOMBUS TILINGS OF A HEXAGON WITH TWO TRIANGLES MISSING ON THE SYMMETRY AXIS THERESIA EISENKÖLBL Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: Theresia.Eisenkoelbl@univie.ac.at
More informationA Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group
A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group Richard P. Stanley Department of Mathematics, Massachusetts Institute of Technology Cambridge,
More informationExcluded permutation matrices and the Stanley Wilf conjecture
Excluded permutation matrices and the Stanley Wilf conjecture Adam Marcus Gábor Tardos November 2003 Abstract This paper examines the extremal problem of how many 1-entries an n n 0 1 matrix can have that
More informationAn Involution for the Gauss Identity
An Involution for the Gauss Identity William Y. C. Chen Center for Combinatorics Nankai University, Tianjin 300071, P. R. China Email: chenstation@yahoo.com Qing-Hu Hou Center for Combinatorics Nankai
More informationON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #A21 ON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS Sergey Kitaev Department of Mathematics, University of Kentucky,
More informationA proof of the Square Paths Conjecture
A proof of the Square Paths Conjecture Emily Sergel Leven October 7, 08 arxiv:60.069v [math.co] Jan 06 Abstract The modified Macdonald polynomials, introduced by Garsia and Haiman (996), have many astounding
More information2 Generating Functions
2 Generating Functions In this part of the course, we re going to introduce algebraic methods for counting and proving combinatorial identities. This is often greatly advantageous over the method of finding
More informationON PARTITION FUNCTIONS OF ANDREWS AND STANLEY
ON PARTITION FUNCTIONS OF ANDREWS AND STANLEY AE JA YEE Abstract. G. E. Andrews has established a refinement of the generating function for partitions π according to the numbers O(π) and O(π ) of odd parts
More informationErdős-Ko-Rado theorems on the weak Bruhat lattice
Erdős-Ko-Rado theorems on the weak Bruhat lattice Susanna Fishel, Glenn Hurlbert, Vikram Kamat, Karen Meagher December 14, 2018 Abstract Let L = (X, ) be a lattice. For P X we say that P is t-intersecting
More informationSYMMETRIC CHAIN DECOMPOSITION FOR CYCLIC QUOTIENTS OF BOOLEAN ALGEBRAS AND RELATION TO CYCLIC CRYSTALS
SYMMETRIC CHAIN DECOMPOSITION FOR CYCLIC QUOTIENTS OF BOOLEAN ALGEBRAS AND RELATION TO CYCLIC CRYSTALS PATRICIA HERSH AND ANNE SCHILLING Abstract. The quotient of a Boolean algebra by a cyclic group is
More informationCIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December Points Possible
Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December 2016 Question Points Possible Points Received 1 12 2 14 3 14 4 12 5 16 6 16 7 16 Total 100 Instructions: 1. This exam
More informationMaximum union-free subfamilies
Maximum union-free subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called
More informationWelsh s problem on the number of bases of matroids
Welsh s problem on the number of bases of matroids Edward S. T. Fan 1 and Tony W. H. Wong 2 1 Department of Mathematics, California Institute of Technology 2 Department of Mathematics, Kutztown University
More informationarxiv: v2 [math.rt] 16 Mar 2018
THE COXETER TRANSFORMATION ON COMINUSCULE POSETS EMINE YILDIRIM arxiv:1710.10632v2 [math.rt] 16 Mar 2018 Abstract. Let J(C) be the poset of order ideals of a cominuscule poset C where C comes from two
More informationEnumerative Combinatorics with Fillings of Polyominoes
Enumerative Combinatorics with Fillings of Polyominoes Catherine Yan Texas A&M Univesrity GSU, October, 204 2 Outline. Symmetry of the longest chains Subsequences in permutations and words Crossings and
More informationALL NORMAL EXTENSIONS OF S5-SQUARED ARE FINITELY AXIOMATIZABLE
ALL NORMAL EXTENSIONS OF S5-SQUARED ARE FINITELY AXIOMATIZABLE Nick Bezhanishvili and Ian Hodkinson Abstract We prove that every normal extension of the bi-modal system S5 2 is finitely axiomatizable and
More informationEnumerative and Algebraic Combinatorics of OEIS A071356
Enumerative and Algebraic Combinatorics of OEIS A071356 Chetak Hossain Department of Matematics North Carolina State University July 9, 2018 Chetak Hossain (NCSU) Combinatorics of OEIS A071356 July 9,
More informationd-regular SET PARTITIONS AND ROOK PLACEMENTS
Séminaire Lotharingien de Combinatoire 62 (2009), Article B62a d-egula SET PATITIONS AND OOK PLACEMENTS ANISSE KASAOUI Université de Lyon; Université Claude Bernard Lyon 1 Institut Camille Jordan CNS UM
More informationA Simple Proof of the Aztec Diamond Theorem
A Simple Proof of the Aztec Diamond Theorem Sen-Peng Eu Department of Applied Mathematics National University of Kaohsiung Kaohsiung 811, Taiwan, ROC speu@nuk.edu.tw Tung-Shan Fu Mathematics Faculty National
More informationPARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS
PARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS GEORGE E. ANDREWS, MATTHIAS BECK, AND NEVILLE ROBBINS Abstract. We study the number p(n, t) of partitions of n with difference t between
More informationA Formula for the Specialization of Skew Schur Functions
A Formula for the Specialization of Skew Schur Functions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher
More informationq-pell Sequences and Two Identities of V. A. Lebesgue
-Pell Seuences and Two Identities of V. A. Lebesgue José Plínio O. Santos IMECC, UNICAMP C.P. 6065, 13081-970, Campinas, Sao Paulo, Brazil Andrew V. Sills Department of Mathematics, Pennsylvania State
More informationThe topology of restricted partition posets
The topology of restricted partition posets Richard Ehrenborg, Jiyoon Jung To cite this version: Richard Ehrenborg, Jiyoon Jung. The topology of restricted partition posets. Bousquet-Mélou, Mireille and
More informationInteger Partitions With Even Parts Below Odd Parts and the Mock Theta Functions
Integer Partitions With Even Parts Below Odd Parts and the Mock Theta Functions by George E. Andrews Key Words: Partitions, mock theta functions, crank AMS Classification Numbers: P84, P83, P8, 33D5 Abstract
More informationINEQUALITIES OF SYMMETRIC FUNCTIONS. 1. Introduction to Symmetric Functions [?] Definition 1.1. A symmetric function in n variables is a function, f,
INEQUALITIES OF SMMETRIC FUNCTIONS JONATHAN D. LIMA Abstract. We prove several symmetric function inequalities and conjecture a partially proved comprehensive theorem. We also introduce the condition of
More informationarxiv: v1 [math.co] 2 Dec 2008
An algorithmic Littlewood-Richardson rule arxiv:08.0435v [math.co] Dec 008 Ricky Ini Liu Massachusetts Institute of Technology Cambridge, Massachusetts riliu@math.mit.edu June, 03 Abstract We introduce
More informationGENERALIZATION OF THE SIGN REVERSING INVOLUTION ON THE SPECIAL RIM HOOK TABLEAUX. Jaejin Lee
Korean J. Math. 8 (00), No., pp. 89 98 GENERALIZATION OF THE SIGN REVERSING INVOLUTION ON THE SPECIAL RIM HOOK TABLEAUX Jaejin Lee Abstract. Eğecioğlu and Remmel [] gave a combinatorial interpretation
More informationCyclic Derangements. Sami H. Assaf. Department of Mathematics MIT, Cambridge, MA 02139, USA
Cyclic Derangements Sami H. Assaf Department of Mathematics MIT, Cambridge, MA 02139, USA sassaf@math.mit.edu Submitted: Apr 16, 2010; Accepted: Oct 26, 2010; Published: Dec 3, 2010 Mathematics Subject
More informationACYCLIC DIGRAPHS GIVING RISE TO COMPLETE INTERSECTIONS
ACYCLIC DIGRAPHS GIVING RISE TO COMPLETE INTERSECTIONS WALTER D. MORRIS, JR. ABSTRACT. We call a directed acyclic graph a CIdigraph if a certain affine semigroup ring defined by it is a complete intersection.
More informationGeneralized Akiyama-Tanigawa Algorithm for Hypersums of Powers of Integers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 16 (2013, Article 1332 Generalized Aiyama-Tanigawa Algorithm for Hypersums of Powers of Integers José Luis Cereceda Distrito Telefónica, Edificio Este
More informationNotation Index. gcd(a, b) (greatest common divisor) NT-16
Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 6: Counting
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 39 Chapter Summary The Basics
More informationEquality of P-partition Generating Functions
Bucknell University Bucknell Digital Commons Honors Theses Student Theses 2011 Equality of P-partition Generating Functions Ryan Ward Bucknell University Follow this and additional works at: https://digitalcommons.bucknell.edu/honors_theses
More informationarxiv: v1 [math.co] 29 Apr 2013
Cyclic permutations realized by signed shifts Kassie Archer and Sergi Elizalde arxiv:1304.7790v1 [math.co] 9 Apr 013 Abstract The periodic (ordinal) patterns of a map are the permutations realized by the
More informationON WEAK CHROMATIC POLYNOMIALS OF MIXED GRAPHS
ON WEAK CHROMATIC POLYNOMIALS OF MIXED GRAPHS MATTHIAS BECK, DANIEL BLADO, JOSEPH CRAWFORD, TAÏNA JEAN-LOUIS, AND MICHAEL YOUNG Abstract. A mixed graph is a graph with directed edges, called arcs, and
More informationORBIT-HOMOGENEITY IN PERMUTATION GROUPS
Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000 ORBIT-HOMOGENEITY IN PERMUTATION GROUPS PETER J. CAMERON and ALEXANDER W. DENT Abstract This paper introduces the
More informationEIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND
EIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND FRANCO SALIOLA Abstract. We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups
More information= i 0. a i q i. (1 aq i ).
SIEVED PARTITIO FUCTIOS AD Q-BIOMIAL COEFFICIETS Fran Garvan* and Dennis Stanton** Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved
More information