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A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k. 8

%I

%S 1,1,4,1,6,19,1,9,39,116,1,12,69,260,751,1,16,119,560,1955,5552,1,20,

%T 189,1100,4615,15372,43219,1,25,294,2090,10460,40677,131131,366088,1,

%U 30,434,3740,22220,100562,370909,1168008,3245311,1,36,630,6512,45628,239316,1007083,3570240,11042199,30569012,1,42,882,10868,89420,541926,2596573,10347864,35587071,108535130,299662672,1,49,1218,17732,170340,1188341,6466159,28915056,110426979,370661885,1117689232,3079276708

%N a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.

%C Maximal element can be any integer, but is chosen here to be <=n.

%H Alois P. Heinz, <a href="/A191714/b191714.txt">Rows n = 1..44, flattened</a>

%H R. Stanley, <a href="http://www-math.mit.edu/~rstan/transparencies/hooks.pdf">Hook Lengths and Contents</a>

%e For n=3 and k=2 the SSYT are

%e par= {3} SSYT= {{1, 1, 1}}, {{2, 1, 1}}, {{2, 2, 1}}, {{2, 2, 2}}

%e par= {2,1} SSYT= {{2, 1}, {1}}, {{2, 2}, {1}}

%e par= {1,1,1} SSYT= none

%e counts 4+2+0 = 6 = a(3,2).

%e Table begins:

%e 1;

%e 1, 4;

%e 1, 6, 19;

%e 1, 9, 39, 116;

%e 1, 12, 69, 260, 751;

%e 1, 16, 119, 560, 1955, 5552;

%e 1, 20, 189, 1100, 4615, 15372, 43219; ...

%t Needs["Combinatorica`"];

%t hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[#1], Max[p]] &) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] &) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] &) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, -1] - 1];

%t content[(p_)?PartitionQ]:= Block[{le= Max[p], ferr =(PadLeft[1+ 0*Range[#1], Max[p]]&) /@ p}, DeleteCases[ MapIndexed[-le+ Range[le,1,-1]- #1- Tr[#2]&, 0*ferr]*ferr,0,-1]+ le];

%t stanley[(p_)?PartitionQ, t_Integer] := Times @@ ((t + Flatten[content[p]])/Flatten[hooklength[p]]);

%t Table[Tr[ stanley[#,k] &/@ Partitions[n] ] , {n,12}, {k,n}]

%Y Cf. A102539, A210391.

%Y Main diagonal gives A209673.

%K nonn,tabl

%O 1,3

%A _Wouter Meeussen_, Jun 12 2011

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Last modified April 19 15:00 EDT 2021. Contains 343116 sequences. (Running on oeis4.)