A309897 - OEIS

%I #24 Mar 17 2020 11:53:49

%S 0,0,0,0,0,0,0,1,2,3,6,9,13,22,33,51,74,104,142,194,261,351,464,616,

%T 802,1047,1344,1716,2186,2766,3473,4367,5448,6774,8375,10329,12685,

%U 15553,18982,23098,28046,33966,40976,49381,59301,71095,85017,101491,120859

%N Number of not unique partition coefficients of n.

%C 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).

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

%e 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.

%e 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].

%p h := proc(n,k) option remember;

%p if n = 0 then return [1] elif k < 1 then return [] fi;

%p [h(n, k-1)[], seq(map(x -> x*k!^j, h(n-k*j, k-1))[], j=1..n/k)] end:

%p A309897 := proc(n) h(n, n); nops(%) - nops(convert(%, set)) end:

%p seq(A309897(n), n=0..48);

%o (SageMath)

%o def A309897(n):

%o     P = Partitions(n)

%o     M = set(multinomial(x) for x in P)

%o     return P.cardinality() - len(M)

%o [A309897(n) for n in range(29)]

%Y Cf. A000041, A070289.

%K nonn

%O 0,9

%A _Peter Luschny_, Sep 06 2019