OFFSET
0,8
COMMENTS
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.
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Oct 02 2024
STATUS
approved