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A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.

%I A094504 #25 Feb 23 2025 11:23:57
%S A094504 1,3,1,6,3,1,13,9,3,1,24,22,9,3,1,48,54,25,9,3,1,86,120,63,25,9,3,1,
%T A094504 160,267,153,66,25,9,3,1,282,559,357,162,66,25,9,3,1,500,1158,805,390,
%U A094504 165,66,25,9,3,1,859,2314,1761,898,399,165,66,25,9,3,1,1479,4559,3761,2025,931,402,165,66,25,9,3,1
%N A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.
%C A094504 First column equals the number of plane partitions of n, corresponding to the 'single layer' solid partitions.
%C A094504 Rows read backward tend to limiting sequence 1, 3, 9, 25, 66, 165, 402, ... A096322.
%H A094504 Wouter Meeussen, <a href="/A094504/b094504.txt">Table of n, a(n) for n = 1..136</a> (16 rows)
%H A094504 Wouter Meeussen, <a href="/A094504/a094504_1.txt">Mma functions for plane and solid partitions</a>
%H A094504 Wouter Meeussen, <a href="/A094504/a094504_3.txt">Mma functions for plane and solid partitions</a>
%F A094504 Finding a G.f. for the solid partitions is an open problem.
%e A094504 T(5,3) = 9 since these 9 solid partitions are [{{3}},{{1}},{{1}}], [{{2,1}},{{1}},{{1}}], [{{1,1,1}},{{1}},{{1}}], [{{2},{1}},{{1}},{{1}}], [{{1,1},{1}},{{1}},{{1}}], [{{1},{1},{1}},{{1}},{{1}}], [{{2}},{{2}},{{1}}], [{{1,1}},{{1,1}},{{1}}], [{{1},{1}},{{1},{1}},{{1}}].
%e A094504 Triangle begins:
%e A094504    1;
%e A094504    3,  1;
%e A094504    6,  3,  1;
%e A094504   13,  9,  3, 1;
%e A094504   24, 22,  9, 3, 1;
%e A094504   48, 54, 25, 9, 3, 1;
%e A094504   ...
%t A094504 (* uses "Mma functions for plane and solid partitions" also used in A090984, A089924 *)
%t A094504  Table[Length/@Split[Sort[Length/@Flatten[solidformBTK/@Partitions[n]]]], {n, 16}]
%Y A094504 Cf. A000293, A090984, A089924, A096322.
%K A094504 nonn,tabl
%O A094504 1,2
%A A094504 _Wouter Meeussen_, Jun 05 2004
%E A094504 Renewed linked Mma program file._Wouter Meeussen_, Feb 20 2025