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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.

%I A191714 #26 Nov 17 2018 16:29:45
%S A191714 1,1,4,1,6,19,1,9,39,116,1,12,69,260,751,1,16,119,560,1955,5552,1,20,
%T A191714 189,1100,4615,15372,43219,1,25,294,2090,10460,40677,131131,366088,1,
%U A191714 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 A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.
%C A191714 Maximal element can be any integer, but is chosen here to be <=n.
%H A191714 Alois P. Heinz, <a href="/A191714/b191714.txt">Rows n = 1..44, flattened</a>
%H A191714 R. Stanley, <a href="http://www-math.mit.edu/~rstan/transparencies/hooks.pdf">Hook Lengths and Contents</a>
%e A191714 For n=3 and k=2 the SSYT are
%e A191714 par= {3}     SSYT= {{1, 1, 1}}, {{2, 1, 1}}, {{2, 2, 1}}, {{2, 2, 2}}
%e A191714 par= {2,1}   SSYT= {{2, 1}, {1}}, {{2, 2}, {1}}
%e A191714 par= {1,1,1} SSYT= none
%e A191714 counts 4+2+0 = 6 = a(3,2).
%e A191714 Table begins:
%e A191714   1;
%e A191714   1,  4;
%e A191714   1,  6,  19;
%e A191714   1,  9,  39,  116;
%e A191714   1, 12,  69,  260,  751;
%e A191714   1, 16, 119,  560, 1955,  5552;
%e A191714   1, 20, 189, 1100, 4615, 15372, 43219; ...
%t A191714 Needs["Combinatorica`"];
%t A191714 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 A191714 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 A191714 stanley[(p_)?PartitionQ, t_Integer] := Times @@ ((t + Flatten[content[p]])/Flatten[hooklength[p]]);
%t A191714 Table[Tr[ stanley[#,k]  &/@ Partitions[n] ] , {n,12}, {k,n}]
%Y A191714 Cf. A102539, A210391.
%Y A191714 Main diagonal gives A209673.
%K A191714 nonn,tabl
%O A191714 1,3
%A A191714 _Wouter Meeussen_, Jun 12 2011