Restricted partition pairs

William H Burge

doi:10.1016/0097-3165(93)90057-f

Untrodden pathways in the theory of the restricted partition function p(n,N)

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2021.105423 ◽

2021 ◽

Vol 180 ◽

pp. 105423

Author(s):

Atul Dixit ◽

Pramod Eyyunni ◽

Bibekananda Maji ◽

Garima Sood

Keyword(s):

Partition Function ◽

Restricted Partition

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Restricted partition functions and identities for degrees of syzygies in numerical semigroups

The Ramanujan Journal ◽

10.1007/s11139-017-9904-7 ◽

2017 ◽

Vol 43 (3) ◽

pp. 465-491 ◽

Cited By ~ 1

Author(s):

Leonid G. Fel

Keyword(s):

Partition Functions ◽

Numerical Semigroups ◽

Restricted Partition

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On the Number of Partition Weights with Kostka Multiplicity One

The Electronic Journal of Combinatorics ◽

10.37236/2574 ◽

2012 ◽

Vol 19 (4) ◽

Author(s):

Zachary Gates ◽

Brian Goldman ◽

C. Ryan Vinroot

Keyword(s):

Positive Integer ◽

Ordered Set ◽

Unitary Groups ◽

Character Theory ◽

Partition Functions ◽

Young Tableaux ◽

Multiplicity One ◽

Restricted Partition ◽

Finite Unitary Groups ◽

Kostka Number

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$. Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$. We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$. We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight. For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$. Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

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