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

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

Author

Peter Luschny, Sep 06 2019

Keywords

• nonn

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.

Examples

			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).
		

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.

Crossrefs

Cf. A036038, A309897, A325472.

Programs

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)