A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

Breuer, Felix; Kronholm, Brandt

doi:10.1007/s40993-015-0033-3

Cited by 9 publications

(10 citation statements)

References 19 publications

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“…G(q) is given in Section 3 in the statement of Lemma 13. Identities similar to Theorem 3 have been explored [3], [2], and [5].…”

Section: Introductionmentioning

confidence: 99%

A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

Larsen

2021

Electron. J. Combin.

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We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity. We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

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“…G(q) is given in Section 3 in the statement of Lemma 13. Identities similar to Theorem 3 have been explored [3], [2], and [5].…”

Section: Introductionmentioning

confidence: 99%

A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

Larsen

2021

Electron. J. Combin.

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“…Interested readers are invited to examine [5], [11], and [12] for other studies on bounded differences between largest and smallest parts. Section 2 has a short repertoire of basic hypergeometric identities that will be referred to later.…”

Section: Introductionmentioning

confidence: 99%

Some Elementary Partition Inequalities and Their Implications

Berkovich

Uncu

2019

Ann. Comb.

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We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for L ≥ 1, the number of partitions with l − s ≤ L and s = 1 is greater than the number of partitions with l − s ≤ L and s > 1. Here l and s are the largest part and the smallest part of the partition, respectively. Date

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“…Later, Breuer and Kronholm [3] studied the numberp(n, t) of partitions of n with the difference between largest and smallest parts bounded by t and obtained…”

Section: Introductionmentioning

confidence: 99%

Partitions With an Arbitrary Number of Specified Distances

Lin

2019

Bull. Aust. Math. Soc.

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For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

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A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

Cited by 9 publications

References 19 publications

A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

Some Elementary Partition Inequalities and Their Implications

Partitions With an Arbitrary Number of Specified Distances

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