A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Author

Pontus von Brömssen, Sep 22 2024

Keywords

• nonn

Comments

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

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

			a(6) = 3, because 6 can be written as a multinomial coefficient in 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.