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 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 On a Reciprocity Law for Finite Multiple Zeta Values

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Markus Kuba ◽  
Helmut Prodinger
Keyword(s):  
Reciprocity Relation ◽  
Weighted Sums ◽  
Combinatorial Proof ◽  
Multiple Zeta Values ◽  
Reciprocity Law ◽  
Multiple Zeta ◽  
Reciprocity Relations ◽  
Zeta Values ◽  
Identity Based ◽  
Partial Fraction Decomposition

It was shown by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.

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.

SIBSC: Separable Identity-Based Signcryption for Resource-Constrained Devices

Informatica ◽  
2017 ◽  
Vol 28 (1) ◽  
pp. 193-214 ◽  
Author(s):  
Tung-Tso Tsai ◽  
Sen-Shan Huang ◽  
Yuh-Min Tseng
Keyword(s):  
Resource Constrained ◽  
Identity Based ◽  
Resource Constrained Devices ◽  
Constrained Devices

Anonymous Hierarchical Identity-Based Encryption with Short Ciphertexts

2011 ◽  
Vol E94-A (1) ◽  
pp. 45-56 ◽  
Author(s):  
Jae Hong SEO ◽  
Tetsutaro KOBAYASHI ◽  
Miyako OHKUBO ◽  
Koutarou SUZUKI
Keyword(s):  
Identity Based Encryption ◽  
Identity Based
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