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A325306
Numbers which are represented by more than one partition of the same integer.
7
56, 210, 504, 1260, 1365, 1680, 1716, 2520, 5040, 7560, 9240, 13860, 15120, 17550, 21840, 24024, 25200, 25740, 27720, 30030, 42504, 43680, 55440, 60060, 69300, 72072, 75600, 77520, 83160, 110880, 120120, 151200, 154440, 166320, 168168, 180180, 185640, 203490
OFFSET
1,1
COMMENTS
We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition and n = p1 + p2 + ... .
We say 'n is represented by p' if n is the partition coefficient of p.
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..10000
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, arXiv:math/0509470 [math.CO], 2005.
EXAMPLE
56 is in this list because it is represented by [5, 3] and [6, 1, 1].
210 is in this list because it is represented by [3, 2, 2] and [4, 1, 1, 1].
These are 'irreducible pairs' of partitions in the terminology of Andrews et al.
Note that the terms can derive from different integers. For instance 27720 is represented by [6, 2, 1, 1, 1] and [5, 3, 2, 1] (partitions of 11) and also by [6, 4, 1, 1] and [5, 4, 3] (partitions of 12).
PROG
(SageMath)
from collections import Counter
def A325306_list(n):
res = []
for k in range(2*n):
L = A309897(k)
d = Counter(L)
res += [j for j, v in d.items() if v > 1]
return sorted(Set(res))[:n]
A325306_list(20)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Sep 06 2019
STATUS
approved