A234968 - OEIS

%I #27 Nov 05 2024 05:40:33

%S 1,2,3,3,3,5,5,5,6,7,5,9,6,9,13,11,7,16,14,14,16,19,14,23,24,21,27,32,

%T 21,39,39,32,38,51,45,56,60,51,62,87,61,82,101,83,98,129,104,120,152,

%U 137,145,196,157,178,248,207,209,293,248,275,353,310,325,441,388,389,528,471,463,656,573,567,766,696,691,934

%N Number of totally symmetric partitions of n of any dimension.

%C a(n) is the sum over d from 1 to infinity of the number of totally symmetric d-dimensional Ferrers diagrams with n nodes.

%C A d-dimensional Ferrers diagram is totally symmetric if and only if whenever X=(x1,x2,...,xd) is a node, then so are all nodes which can be specified by permuting the coordinates of X.

%C Since a(1)=oo, the sequence above begins on n=2. All other terms are finite.

%H Graham H. Hawkes, <a href="/A234968/b234968.txt">Table of n, a(n) for n = 2..90</a>

%H Graham H. Hawkes, <a href="/A234968/a234968_2.txt">Table of TS FD for dim 1...7</a>

%e a(1)=oo because for each dimension, d, the trivial Ferrers diagram given by the single node (1,1,1,...,1) is a totally symmetric d-dimensional partition of 1.

%e For n > 2, a(n) < oo. This means that for n > 2, there are at most a finite number of dimensions, d, for which the number of totally symmetric d-dimensional partitions of n is nonzero (and that for any dimension, d, there are at most a finite number of totally symmetric d-dimensional partitions of n).

%e a(2)=1. Indeed the only totally symmetric partition of 2 occurs in dimension 1. The corresponding 1-dimensional totally symmetric Ferrers diagram (TS FD) is given by the following two nodes (specified by the 1-dimensional coordinates): (2) and (1).

%e a(8)=5.

%e There is one 1-dimensional TS FD of 8:

%e   {(8),(7),(6),(5),(4),(3),(2),(1)}

%e There are two 2-dimensional TS FD of 8:

%e   {(3,2),(2,3),(3,1),(2,2),(1,3),(2,1),(1,2),(1,1)} and

%e   {(4,1),(1,4),(3,1),(2,2),(1,3),(2,1),(1,2),(1,1)}

%e There is one 3-dimensional TS FD of 8:

%e   {(2,2,2),(2,2,1),(2,1,2),(1,2,2),(2,1,1),(1,2,1),(1,1,2),(1,1,1)}

%e There is one 7-dimensional TS FD of 8:

%e   {(2,1,1,1,1,1,1),(1,2,1,1,1,1,1),(1,1,2,1,1,1,1),(1,1,1,2,1,1,1),(1,1,1,1,2,1,1),(1,1,1,1,1,2,1),(1,1,1,1,1,1,2),(1,1,1,1,1,1,1)}

%e There are no TS FD of 8 of any other dimension. Hence a(8)=1+2+1+1=5.

%e a(72)=573

%e The TS FD of 72 are:

%e   Dim 1: 1

%e   Dim 2: 471

%e   Dim 3: 85

%e   Dim 4: 11

%e   Dim 5: 3

%e   Dim 6: 1

%e   Dim 71: 1

%e (For n > 1) there is always exactly 1 TS FD of dimension 1 and 1 TS FD of dimension n-1. If n > 2, these two dimensions are not equal, so there must be at least two TS FD. Hence a(n) >= 2 for n > 2.

%Y The number of TS FD of dimensions 2, 3, and 4 are given by sequences A000700, A048141, and A097516 respectively.

%K nonn,nice

%O 2,2

%A _Graham H. Hawkes_, Jan 02 2014