%I #17 Feb 09 2023 22:09:04

%S 1,2,6,32,352,9304,683464,161960220

%N Number of totally symmetric solid partitions which fit in an n X n X n X n box.

%C Also, for n > 0, the number of totally symmetric (n-1)-dimensional partitions which fit in an (n-1)-dimensional box whose sides all have length 5.

%C There is no conjectured formula for a(n).

%C The formula a(n,d) = Product_{i_1=1..n} Product_{i_2=i_1..n} ... Product_{i_d=i_(d-1)..n} (i_1+i_2+...+i_d-d+2)/(i_1+i_2+...+i_d-d+1) gives the number of totally symmetric d-dimensional partitions that fit in a box whose sides all have length n, for d = 1, 2, and 3. For d > 3 this formula fails. In particular, when d=4 it produces the sequence: 1, 2, 6, 32, 352, 9216, 661504, ... rather than the sequence above.

%H Seth Ireland, <a href="https://arxiv.org/abs/2302.02505">A bijection between strongly stable and totally symmetric partitions</a>, arXiv:2302.02505 [math.CO], 2023.

%Y This is the 4-dimensional case. Dimensions 1, 2, and 3 are respectively given by A000027, A000079, and A005157.

%Y Cf. A097516.

%K nonn,hard,more

%O 0,2

%A _Graham H. Hawkes_, Jan 30 2014