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.

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, 18, 19, 20, 23, 24, 26, 27, 30, 33, 37, 40, 43, 49, 52, 57, 64, 68, 76, 79, 87, 93, 99, 109, 116, 125, 135, 143, 157, 171, 191, 206, 223, 238, 254, 276, 291

Offset: 0

Author

Pontus von Brömssen, Oct 02 2024

Keywords

• nonn

Comments

Frequency of the most common number in row n of A036038 (for n >= 1) or A078760.

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.

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.

Examples

			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.
		

Crossrefs

Cf. A036038, A070289, A078760, A376369, A376662, A376663.