The restricted partition lattices

2019 ◽  
Author(s):  
István Mező
Keyword(s):  
Restricted Partition ◽  
Partition Lattices

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

2021 ◽  
Vol 180 ◽  
pp. 105423
Author(s):  
Atul Dixit ◽  
Pramod Eyyunni ◽  
Bibekananda Maji ◽  
Garima Sood
Keyword(s):  
Partition Function ◽  
Restricted Partition

scholarly journals Restricted partition pairs

1993 ◽  
Vol 63 (2) ◽  
pp. 210-222 ◽  
Author(s):  
William H Burge
Keyword(s):  
Restricted Partition

Algebraic Aspects of Partition Lattices

1992 ◽  
pp. 106-122 ◽  
Author(s):  
Ivan Rival ◽  
Miriam Stanford
Keyword(s):  
Partition Lattices

scholarly journals On the Number of Partition Weights with Kostka Multiplicity One

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.

scholarly journals Chains, antichains, and complements in infinite partition lattices

2018 ◽  
Vol 79 (2) ◽  
Author(s):  
James Emil Avery ◽  
Jean-Yves Moyen ◽  
Pavel Růžička ◽  
Jakob Grue Simonsen
Keyword(s):  
Partition Lattices
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