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

0,3

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).

Links

Table of n, a(n) for n=0..66.

Wikipedia, Young tableau

Example

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.

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]}]

Crossrefs

Row sums give A000312.

Row lengths give A000041.

Leftmost column gives A088218.

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

Sequence in context: A212930 A225725 A095327 * A225753 A210725 A048953

Adjacent sequences: A380605 A380606 A380607 * A380612 A380613 A380614

Keyword

nonn,tabf,new

Author

Wouter Meeussen, Jan 28 2025

Status

approved