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A380611 Irregular triangle read by rows: T(r,c) is the product of the number of standard Young tableaux (A117506) and the number of semistandard Young tableaux (A262030) for partitions of r.

%I A380611 #34 Jan 29 2025 22:15:35
%S A380611 1,1,3,1,10,16,1,35,135,40,45,1,126,896,875,756,375,96,1,462,5250,
%T A380611 10206,8400,2450,14336,2800,875,1701,175,1,1716,28512,90552,74250,
%U A380611 65856,257250,48000,74088,55566,102900,8100,10976,5488,288,1,6435,147147,686400,567567,931392,3244032,606375,194040,2910600,1448832,2673000,202125,666792,846720,1029000,491520,19845,24696,65856,14400,441,1
%N A380611 Irregular triangle read by rows: T(r,c) is the product of the number of standard Young tableaux (A117506) and the number of semistandard Young tableaux (A262030) for partitions of r.
%C A380611 Partitions are generated in reverse lexicographic order.
%C A380611 Remark that A262030 uses Abramowitz-Stegun (A-St) order.
%C A380611 Sum of row r equals r^r for r > 0 (Robinson-Schensted correspondence).
%H A380611 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A380611 Triangle begins:
%e A380611     1;
%e A380611     1;
%e A380611     3,    1;
%e A380611    10,   16,     1;
%e A380611    35,  135,    40,   45,    1;
%e A380611   126,  896,   875,  756,  375,    96,    1;
%e A380611   462, 5250, 10206, 8400, 2450, 14336, 2800, 875, 1701, 175, 1;
%e A380611   ...
%e A380611 Fourth row is 1*35, 3*45, 2*20, 3*15, 1*1 with sum 256 = 4^4.
%t A380611 Needs["Combinatorica`"];
%t A380611 hooklength[par_?PartitionQ]:=Table[Count[par,q_/;q>=j]+1-i+par[[i]]-j,{i,Length[par]},{j,par[[i]]}];
%t A380611 countSYT[par_?PartitionQ]:=Tr[par]!/Times@@Flatten[hooklength[par]];
%t A380611 content[par_?PartitionQ]:=Table[j-i,{i,Length[par]},{j,par[[i]]}];
%t A380611 countSSYT[par_?PartitionQ,t_Integer_]:=Times@@((t+Flatten[content[par]])/Flatten[hooklength[par]]);
%t A380611 Table[countSYT[par] countSSYT[par,n],{n,8},{par,IntegerPartitions[n]}]
%Y A380611 Row sums give A000312.
%Y A380611 Row lengths give A000041.
%Y A380611 Leftmost column gives A088218.
%Y A380611 Cf. A047874, A059304, A365643, A262030, A117506.
%K A380611 nonn,tabf
%O A380611 0,3
%A A380611 _Wouter Meeussen_, Jan 28 2025