scholarly journals A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem

10.37236/4561 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
James J.Y. Zhao
Keyword(s):  
Combinatorial Proof ◽  
Partition Theorem ◽  
Bijective Proof

Based on the combinatorial proof of Schur's partition theorem given by Bressoud, and the combinatorial proof of Alladi's partition theorem given by Padmavathamma, Raghavendra and Chandrashekara, we obtain a bijective proof of a partition theorem of Alladi, Andrews and Gordon.

scholarly journals Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Stirling Numbers ◽  
Algebraic Methods ◽  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Long Cycle ◽  
Bijective Proof ◽  
International Audience

International audience We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account. Nous étudions le nombre de permutations $\beta$ de $[N]$ avec $m$ cycles telles que $(1 2 \ldots N) \beta^{-1}$ a un seul cycle. Ces nombres apparaissent en tant que coefficients des monômes linéaires des polynômes de Kerov et de Stanley. À l'aide de méthodes algébriques, D. Zagier a trouvé une connexion inattendue avec les nombres de Stirling de taille $N+1$. Nous présentons ici la première preuve combinatoire de son résultat, en introduisant une nouvelle bijection entre des cartes partitionnées et des arbres épineux. De plus, nous obtenons un résultat plus fin, prenant en compte le type des permutations.

scholarly journals Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Shishuo Fu ◽  
James Sellers
Keyword(s):  
Partition Identities ◽  
Partition Theorem ◽  
Bijective Proof ◽  
International Audience

International audience We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod{6}$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result. Nous revisitons un théorème de partitions d'entiers dû à MacMahon, qui relie les partitions dont chaque part est répétée au moins une fois et celles dont les parts sont congrues à $2, 3, 4, 6 \pmod{6}$, ainsi qu'une généralisation par Andrews et deux autres par Subbarao. Ensuite nous construisons unepreuve bijective unifiée pour tous les quatre théorèmes ci-dessus, et obtenons de plus une généralisation naturelle.

scholarly journals An Involution Proof of the Alladi-Gordon Key Identity for Schur's Partition Theorem

10.37236/2826 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
James J.Y. Zhao
Keyword(s):  
Combinatorial Proof ◽  
Partition Theorem ◽  
Insertion Algorithm ◽  
Identity Based ◽  
Gordon Identity

The Alladi-Gordon identity $\sum_{k=0}^{j}(q^{i-k+1};q)_k\, {j \brack k} q^{(i-k)(j-k)}=1$ plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by establishing an involution on the underlying set of overpartitions.

scholarly journals Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

10.37236/3907 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shishuo Fu ◽  
James Allen Sellers
Keyword(s):  
Partition Identities ◽  
Partition Theorem ◽  
Bijective Proof

We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod 6$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.

scholarly journals Self-Dual Interval Orders and Row-Fishburn Matrices

10.37236/2201 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Sherry H. F. Yan ◽  
Yuexiao Xu
Keyword(s):  
Generating Functions ◽  
Combinatorial Proof ◽  
Interval Orders ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Bijective Proof

Recently, Jelínek derived  that the number of self-dual interval orders of reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof  of the refined relations between self-dual Fishburn matrices and row-Fishburn matrices in answer to a problem proposed by Jelínek.

scholarly journals Euler's Partition Theorem with Upper Bounds on Multiplicities

10.37236/2318 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
William Y.C. Chen ◽  
Ae Ja Yee ◽  
Albert J. W. Zhu
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Combinatorial Proof ◽  
Partition Theorem ◽  
Two Parameters

We show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is bounded by $m$. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For $m=0$, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each even part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is also bounded by $2m+1$. We provide a combinatorial proof as well.

scholarly journals A Combinatorial Proof of a Symmetric $q$-Pfaff-Saalschütz Identity

10.37236/1969 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Elementary Proof ◽  
Combinatorial Proof ◽  
Bijective Proof

We give a bijective proof of a symmetric $q$-identity on $_4\phi_3$ series, which is a symmetric generalization of the famous $q$-Pfaff-Saalschütz identity. An elementary proof of this identity is also given.

scholarly journals A new bijective proof of a partition theorem of K. Alladi

2004 ◽  
Vol 287 (1-3) ◽  
pp. 125-128
Author(s):  
Padmavathamma ◽  
R. Raghavendra ◽  
B.M. Chandrashekara
Keyword(s):  
Partition Theorem ◽  
Bijective Proof
Sign in / Sign up

Export Citation Format

Share Document