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MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares
"... . A signicant portion of MacMahon's famous book \Combinatory Analysis " is devoted to the development of \Partition Analysis" as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMah ..."
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. A signicant portion of MacMahon's famous book \Combinatory Analysis " is devoted to the development of \Partition Analysis" as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon's ideas have not received due attention with the exception of work by Richard Stanley. A long range object of a series of articles is to change this situation by demonstrating the power of MacMahon's method in current combinatorial and partitiontheoretic research. The renaissance of MacMahon's technique partly is due to the fact that it is ideally suited for being supplemented by modern computer algebra methods. In this paper we illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three dierent aspects of combinatorial work: the construction of bijections (for the Rened Lecture Hall Partition Theorem), exploitation of recursive patterns (for Cayley composit...
MacMahon’s Partition Analysis IV: Hypergeometric Multisums
, 1999
"... In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. The object of this paper is to introduce an entirely new application domain for ..."
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Cited by 18 (5 self)
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In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. The object of this paper is to introduce an entirely new application domain for MacMahon’s operator technique. Namely, we show that Partition Analysis can be also used for proving hypergeometric multisum identities. Our examples range from combinatorial sums involving binomial coefficients, harmonic and derangement numbers to multisums which arise in physics and which are related to the KnuthBender theorem.
Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences
 J. Combin. Theory Ser. A
, 2012
"... For a sequence s = (s1,..., sn) of positive integers, an slecture hall partition is an integer sequence λ satisfying 0 ≤ λ1/s1 ≤ λ2/s2 ≤... ≤ λn/sn. In this work, we introduce slecture hall polytopes, sinversion sequences, and relevant statistics on both families. We show that for any sequence s ..."
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For a sequence s = (s1,..., sn) of positive integers, an slecture hall partition is an integer sequence λ satisfying 0 ≤ λ1/s1 ≤ λ2/s2 ≤... ≤ λn/sn. In this work, we introduce slecture hall polytopes, sinversion sequences, and relevant statistics on both families. We show that for any sequence s of positive integers: (i) the h ∗vector of the slecture hall polytope is the ascent polynomial for the associated sinversion sequences; (ii) the ascent polynomials for sinversion sequences generalize the Eulerian polynomials, including a qanalog that tracks a generalization of major index on sinversion sequences; and (iii) the generating function for the slecture hall partitions can be interpreted in terms of a new qanalog of the sEulerian polynomials, which tracks a “lecture hall ” statistic on sinversion sequences. We show how four different statistics are related through the three sfamilies of partitions, polytopes, and inversion sequences. Our approach uses Ehrhart theory to relate the partition theory of lecture hall partitions to their geometry.
MacMahon’s Partition Analysis: The Omega Package
 ARTICLE SUBMITTED TO EUROPEAN JOURNAL OF COMBINATORICS
"... In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis (“Omega Calculus”) as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. The object of this paper is to show that partition analysis is ideally suit ..."
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Cited by 16 (3 self)
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In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis (“Omega Calculus”) as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. The object of this paper is to show that partition analysis is ideally suited for being implemented in computer algebra. To this end we have developed the computer algebra package Omega. In addition to an introduction to basic facts of “Omega Calculus”, we present a number of applications that illustrate the usage of the package.
Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences
, 2009
"... Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n − 1} + t {n − 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by n ..."
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Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n − 1} + t {n − 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by n
Abacus models for parabolic quotients of affine Weyl groups
 J. Algebra
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A Note on Partitions Into Distinct Parts and Odd Parts
, 1999
"... . BousquetMelou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a wellknown result by Euler. We give a di#erent graphical interpretation of the bijection by Sy ..."
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. BousquetMelou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a wellknown result by Euler. We give a di#erent graphical interpretation of the bijection by Sylvester on partitions into distinct parts and partitions into odd parts, and show that the bijection implies the above statement. Keywords: Partitions 1. Introduction A finite nonincreasing sequence of positive integers, # = # 1 # 2 # l , is called an integer partition of n, where n = P l i=1 # i . We follow standard notations in [1]. The length of a partition # is the number of its parts, denoted l(#). Let #, called the size of #, denote the number which # is a partition of. Though each part in a partition is positive, we sometimes allow a zero as a part. Let P d denote the set of all partitions whose parts are all distinct, and let P o denote the set of all partitions whose part...
A Unification of Two Refinements of Euler’s Partition Theorem
, 2009
"... We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the AndrewsOlsson partition identity is used in our combinatorial construction. ..."
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We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the AndrewsOlsson partition identity is used in our combinatorial construction.
Affine partitions and affine Grassmannians
"... We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousque ..."
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We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of BousquetMelouEriksson, ErikssonEriksson and Reiner. In other types the identities appear to be new. For type An, the affine colored partitions form another family of combinatorial objects in bijection with (n+1)core partitions and nbounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young’s lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted. 1