A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

2019 ◽  
Author(s):  
Dennis Eichhorn ◽  
Hayan Nam ◽  
Jaebum Sohn
Keyword(s):  
Pentagonal Number Theorem

scholarly journals Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

scholarly journals Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

10.37236/1796 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
James A. Sellers ◽  
Andrew V. Sills ◽  
Gary L. Mullen
Keyword(s):  
Famous Theorem ◽  
Partition Identities ◽  
Product Identity ◽  
Powers Of 2 ◽  
Pentagonal Number Theorem ◽  
Series Product

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

On the sum of parts with multiplicity at least 2 in all the partitions of n

2020 ◽  
pp. 1-17
Author(s):  
Mircea Merca ◽  
Ae Ja Yee
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Interpretations ◽  
Pentagonal Number Theorem

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

scholarly journals ON A COMBINATORIAL PROOF FOR AN IDENTITY INVOLVING THE TRIANGULAR NUMBERS

2010 ◽  
Vol 83 (1) ◽  
pp. 46-49
Author(s):  
JOSE PLÍNIO O. SANTOS ◽  
ROBSON DA SILVA
Keyword(s):  
Combinatorial Proof ◽  
Triangular Numbers ◽  
Pentagonal Number Theorem

AbstractIn this paper, we present a combinatorial proof for an identity involving the triangular numbers. The proof resembles Franklin’s proof of Euler’s pentagonal number theorem.

Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Chuanan Wei ◽  
Dianxuan Gong
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Jacobi Triple Product Identity ◽  
Liouville’S Theorem

AbstractBy means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity.

scholarly journals A formula for the number of partitions of n in terms of the partial Bell polynomials

2020 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Bell Polynomials ◽  
Pentagonal Number Theorem

We derive a formula for $p(n)$ (the number of partitions of n) in terms of the partial Bell polynomials using Faà di Bruno’s formula and Euler’s pentagonal number theorem.

scholarly journals Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

10.37236/4917 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Louis W. Kolitsch ◽  
Michael Burnette
Keyword(s):  
Partial Sums ◽  
Number Series ◽  
Linear Combinations ◽  
Pentagonal Number Theorem

In 2012 Andrews and Merca gave a new expansion for partial sums of Euler's pentagonal number series and expressed \[\sum_{j=0}^{k-1}(-1)^j(p(n-j(3j+1)/2)-p(n-j(3j+5)/2-1))=(-1)^{k-1}M_k(n)\] where $M_k(n)$ is the number of partitions of $n$ where $k$ is the least integer that does not occur as a part and there are more parts greater than $k$ than there are less than $k$. We will show that $M_k(n)=C_k(n)$ where $C_k(n)$ is the number of partition pairs $(S, U)$ where $S$ is a partition with parts greater than $k$, $U$ is a partition with $k-1$ distinct parts all of which are greater than the smallest part in $S$, and the sum of the parts in $S \cup U$ is $n$. We use partition pairs to determine what is counted by three similar expressions involving linear combinations of pentagonal numbers. Most of the results will be presented analytically and combinatorially.

Euler's Pentagonal Number Theorem

1983 ◽  
Vol 56 (5) ◽  
pp. 279 ◽  
Author(s):  
George E. Andrews
Keyword(s):  
Pentagonal Number Theorem
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