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.

1, 1, 3, 1, 10, 16, 1, 35, 135, 40, 45, 1, 126, 896, 875, 756, 375, 96, 1, 462, 5250, 10206, 8400, 2450, 14336, 2800, 875, 1701, 175, 1, 1716, 28512, 90552, 74250, 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

Offset: 0

Author

Wouter Meeussen, Jan 28 2025

Keywords

• nonn
• tabf

Comments

Partitions are generated in reverse lexicographic order.

Remark that A262030 uses Abramowitz-Stegun (A-St) order.

Sum of row r equals r^r for r > 0 (Robinson-Schensted correspondence).

Examples

			Triangle begins:
    1;
    1;
    3,    1;
   10,   16,     1;
   35,  135,    40,   45,    1;
  126,  896,   875,  756,  375,    96,    1;
  462, 5250, 10206, 8400, 2450, 14336, 2800, 875, 1701, 175, 1;
  ...
Fourth row is 1*35, 3*45, 2*20, 3*15, 1*1 with sum 256 = 4^4.
		

Links

• Wikipedia, Young tableau

Crossrefs

Row sums give A000312.

Row lengths give A000041.

Leftmost column gives A088218.

Cf. A047874, A059304, A365643, A262030, A117506.

Programs

Mathematica

Needs["Combinatorica`"];
hooklength[par_?PartitionQ]:=Table[Count[par,q_/;q>=j]+1-i+par[[i]]-j,{i,Length[par]},{j,par[[i]]}];
countSYT[par_?PartitionQ]:=Tr[par]!/Times@@Flatten[hooklength[par]];
content[par_?PartitionQ]:=Table[j-i,{i,Length[par]},{j,par[[i]]}];
countSSYT[par_?PartitionQ,t_Integer_]:=Times@@((t+Flatten[content[par]])/Flatten[hooklength[par]]);
Table[countSYT[par] countSSYT[par,n],{n,8},{par,IntegerPartitions[n]}]