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Numbers which are represented by more than one partition of the same integer.
7

%I #37 Sep 30 2024 10:47:06

%S 56,210,504,1260,1365,1680,1716,2520,5040,7560,9240,13860,15120,17550,

%T 21840,24024,25200,25740,27720,30030,42504,43680,55440,60060,69300,

%U 72072,75600,77520,83160,110880,120120,151200,154440,166320,168168,180180,185640,203490

%N Numbers which are represented by more than one partition of the same integer.

%C We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition and n = p1 + p2 + ... .

%C We say 'n is represented by p' if n is the partition coefficient of p.

%H Pontus von Brömssen, <a href="/A325306/b325306.txt">Table of n, a(n) for n = 1..10000</a>

%H George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, <a href="http://arxiv.org/abs/math/0509470">On the Number of Distinct Multinomial Coefficients</a>, arXiv:math/0509470 [math.CO], 2005.

%e 56 is in this list because it is represented by [5, 3] and [6, 1, 1].

%e 210 is in this list because it is represented by [3, 2, 2] and [4, 1, 1, 1].

%e These are 'irreducible pairs' of partitions in the terminology of Andrews et al.

%e 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).

%o (SageMath)

%o from collections import Counter

%o def A325306_list(n):

%o res = []

%o for k in range(2*n):

%o L = A309897(k)

%o d = Counter(L)

%o res += [j for j, v in d.items() if v > 1]

%o return sorted(Set(res))[:n]

%o A325306_list(20)

%Y Cf. A036038, A309897, A325472.

%K nonn

%O 1,1

%A _Peter Luschny_, Sep 06 2019