A262030 -id:A262030 - cp's OEIS frontend

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

Wouter Meeussen, Jan 28 2025

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).

			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.

Wikipedia, Young tableau

Row sums give A000312.
Row lengths give A000041.
Leftmost column gives A088218.
Cf. A047874, A059304, A365643, A262030, A117506.

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]}]

A054688 Number of nonnegative integer n X n matrices with sum of elements equal to n; polynomial symmetric functions of matrix of order n.

Original entry on oeis.org

1, 1, 10, 165, 3876, 118755, 4496388, 202927725, 10639125640, 635627275767, 42634215112710, 3172596834321200, 259398433286078100, 23116565732981832150, 2230164446387219893320, 231574204669402103059965, 25751746463640423324267024, 3053419608195531383028424575

Offset: 0

Vladeta Jovovic, Apr 19 2000

Conjecture: a(n) equals the sum of squares of the number of semistandard Young tableaux over all partitions of n. - Wouter Meeussen, Feb 02 2025

E. R. Cavalcanti and M. A. Spohn, On the applicability of mobility metrics for user movement pattern recognition in MANETs, in Proceeding MobiWac '13 Proceedings of the 11th ACM international symposium on Mobility management and wireless access, Pages 123-130, ACM New York, NY, USA 2013, ISBN: 978-1-4503-2355-0 doi:10.1145/2508222.2508228

Alois P. Heinz, Table of n, a(n) for n = 0..337

Cf. A001700, A007716, A262030.

a:= n-> binomial(n*(n+1)-1, n):
seq(a(n), n=0..17);  # Alois P. Heinz, Oct 22 2021

Table[Binomial[n^2 + n - 1, n], {n, 0, 17}] (* Michael De Vlieger, Oct 05 2017 *)

a(n) = binomial(n^2+n-1, n); \\ Altug Alkan, Oct 03 2017

a(n) = C(n^2+n-1, n).
a(n) = [x^n] 1/(1 - x)^(n^2). - Ilya Gutkovskiy, Oct 03 2017
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 06 2025

a(15) corrected by Ilya Gutkovskiy, Oct 03 2017

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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.

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A054688 Number of nonnegative integer n X n matrices with sum of elements equal to n; polynomial symmetric functions of matrix of order n.

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