cp's OEIS Frontend

A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

%I A376661 #5 Oct 02 2024 14:23:03
%S A376661 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,6,6,7,8,9,11,11,13,13,14,15,16,
%T A376661 18,19,20,23,24,26,27,30,33,37,40,43,49,52,57,64,68,76,79,87,93,99,
%U A376661 109,116,125,135,143,157,171,191,206,223,238,254,276,291
%N A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.
%C A376661 Frequency of the most common number in row n of A036038 (for n >= 1) or A078760.
%C A376661 The sequence is nondecreasing, because a set of partitions of n-1 with a common multinomial coefficient can be extended to a set of partitions of n with a common multinomial coefficient by adding a unit part to each partition. It appears that a(n) > a(n-1) for n >= 28.
%C A376661 The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.
%e A376661 For n = 7, the only number that appears more than once in row 7 of A036038 is 210, which appears twice: 210 = 7!/(2!*2!*3!) = 7!/(1!*1!*1!*4!). Hence, a(7) = 2.
%Y A376661 Cf. A036038, A070289, A078760, A376369, A376662, A376663.
%K A376661 nonn
%O A376661 0,8
%A A376661 _Pontus von Brömssen_, Oct 02 2024