A Combinatorial Proof of Schur's 1926 Partition Theorem

1980 ◽  
Vol 79 (2) ◽  
pp. 338 ◽  
Author(s):  
David M. Bressoud
Keyword(s):  
Combinatorial Proof ◽  
Partition Theorem

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 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 Euler's partition theorem and the combinatorics of ℓ-sequences

2008 ◽  
Vol 115 (6) ◽  
pp. 967-996 ◽  
Author(s):  
Carla D. Savage ◽  
Ae Ja Yee
Keyword(s):  
Partition Theorem

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

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