A309897 Number of not unique partition coefficients of n.

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 6, 9, 13, 22, 33, 51, 74, 104, 142, 194, 261, 351, 464, 616, 802, 1047, 1344, 1716, 2186, 2766, 3473, 4367, 5448, 6774, 8375, 10329, 12685, 15553, 18982, 23098, 28046, 33966, 40976, 49381, 59301, 71095, 85017, 101491, 120859

Offset

0,9

Comments

We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition, n = p1 + p2 + ... and denote it by P(n, p).

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A000041(n) - A070289(n).

Example

a(7) = 1 because the partition coefficients of 7 are [1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040], P(7, [3, 2, 2]) = P(7, [4, 1, 1, 1]) = 210 and all other partition coefficients are unique.
We say that two partitions of n are multinomial-equivalent if they have the same partition coefficient. For instance [6, 2, 2, 1, 1] ~ [5, 4, 1, 1, 1] ~ [5, 3, 2, 2] and [6, 4, 1, 1, 1, 1, 1] ~ [6, 3, 2, 2, 1, 1] ~ [5, 4, 3, 1, 1, 1] ~ [5, 3, 3, 2, 2].

Maple

h := proc(n, k) option remember;
if n = 0 then return [1] elif k < 1 then return [] fi;
[h(n, k-1)[], seq(map(x -> x*k!^j, h(n-k*j, k-1))[], j=1..n/k)] end:
A309897 := proc(n) h(n, n); nops(%) - nops(convert(%, set)) end:
seq(A309897(n), n=0..48);

Prog

(SageMath)
def A309897(n):
    P = Partitions(n)
    M = set(multinomial(x) for x in P)
    return P.cardinality() - len(M)
[A309897(n) for n in range(29)]

Crossrefs

Cf. A000041, A070289.

Sequence in context: A357619 A360362 A280422 * A128955 A215523 A306247

Adjacent sequences: A309894 A309895 A309896 * A309898 A309899 A309900

Keyword

nonn

Author

Peter Luschny, Sep 06 2019

Status

approved