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

1,1

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

Table of n, a(n) for n=1..38.

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

Cf. A036038, A309897, A325472.

Sequence in context: A241611 A179403 A158481 * A193428 A179757 A233989

Adjacent sequences: A325303 A325304 A325305 * A325307 A325308 A325309

Keyword

nonn

Author

Peter Luschny, Sep 06 2019

Status

approved