cp's OEIS Frontend

A325306 Numbers which are represented by more than one partition of the same integer.

%I A325306 #37 Sep 30 2024 10:47:06
%S A325306 56,210,504,1260,1365,1680,1716,2520,5040,7560,9240,13860,15120,17550,
%T A325306 21840,24024,25200,25740,27720,30030,42504,43680,55440,60060,69300,
%U A325306 72072,75600,77520,83160,110880,120120,151200,154440,166320,168168,180180,185640,203490
%N A325306 Numbers which are represented by more than one partition of the same integer.
%C A325306 We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition and n = p1 + p2 + ... .
%C A325306 We say 'n is represented by p' if n is the partition coefficient of p.
%H A325306 Pontus von Brömssen, <a href="/A325306/b325306.txt">Table of n, a(n) for n = 1..10000</a>
%H A325306 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 A325306 56 is in this list because it is represented by [5, 3] and [6, 1, 1].
%e A325306 210 is in this list because it is represented by [3, 2, 2] and [4, 1, 1, 1].
%e A325306 These are 'irreducible pairs' of partitions in the terminology of Andrews et al.
%e A325306 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 A325306 (SageMath)
%o A325306 from collections import Counter
%o A325306 def A325306_list(n):
%o A325306     res = []
%o A325306     for k in range(2*n):
%o A325306         L = A309897(k)
%o A325306         d = Counter(L)
%o A325306         res += [j for j, v in d.items() if v > 1]
%o A325306     return sorted(Set(res))[:n]
%o A325306 A325306_list(20)
%Y A325306 Cf. A036038, A309897, A325472.
%K A325306 nonn
%O A325306 1,1
%A A325306 _Peter Luschny_, Sep 06 2019