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

2017 ◽  
Vol 43 (3) ◽  
pp. 465-491 ◽  
Author(s):  
Leonid G. Fel
Keyword(s):  
Partition Functions ◽  
Numerical Semigroups ◽  
Restricted Partition

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 Some Restricted Partition Functions

1993 ◽  
Vol 45 (2) ◽  
pp. 228-240 ◽  
Author(s):  
P. Borwein
Keyword(s):  
Partition Functions ◽  
Restricted Partition

On congruence properties of the restricted partition functions $$p_{\omega }(n,k)$$ p ω ( n , k ) and $$p_{\nu }(n,k)$$ p ν ( n , k )

2018 ◽  
Vol 49 (1) ◽  
pp. 105-113
Author(s):  
Robson da Silva ◽  
Kelvin Souza de Oliveira ◽  
Almir Cunha da Graça Neto
Keyword(s):  
Partition Functions ◽  
Restricted Partition ◽  
Congruence Properties

scholarly journals Some Restricted Partition Functions: Congruences Modulo 5

1969 ◽  
Vol 9 (3-4) ◽  
pp. 424-432 ◽  
Author(s):  
D. B. Lahiri
Keyword(s):  
Partition Functions ◽  
Restricted Partition ◽  
Image Position ◽  
Almost All

Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence,for example, is famous [2; 3]. Some progress has been made since then; it is known that the congruence,has an infinitude of solutions for any arbitrary value of r [4]. This is a somewhat weak relation, and one would have liked to obtain, if possible, stronger results of the type,for ‘almost all’ values of n, which in its turn is derivable from another stronger relation, viz.,also established by Ramanujan [2], where r(n) is Ramanujan's function defined by

scholarly journals Some restricted partition functions: Congruences modulo 3

1970 ◽  
Vol 11 (1) ◽  
pp. 82-90 ◽  
Author(s):  
D. B. Lahiri
Keyword(s):  
Partition Functions ◽  
General Category ◽  
Previous Communication ◽  
Restricted Partition ◽  
Image Position ◽  
Congruence Properties

In a previous communication [5] the author has dealt with the congruence properties of some restricted partition functions. The general category of such functions may be denoted.

scholarly journals Estimates of five restricted partition functions that are quasi polynomials

2014 ◽  
Vol 5 (1) ◽  
pp. 1-11
Author(s):  
A. David Christopher ◽  
M. Davamani Christober
Keyword(s):  
Partition Functions ◽  
Restricted Partition
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