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%I #34 Nov 17 2018 16:38:58
%S 1,1,0,1,1,0,1,2,1,0,1,3,4,1,0,1,4,9,6,1,0,1,5,16,19,9,1,0,1,6,25,44,
%T 39,12,1,0,1,7,36,85,116,69,16,1,0,1,8,49,146,275,260,119,20,1,0,1,9,
%U 64,231,561,751,560,189,25,1,0
%N Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A210391/b210391.txt">Antidiagonals n = 0..140, flattened</a>
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/SemistandardTableaux">Semistandard Young tableaux</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).
%F A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - _Alois P. Heinz_, Apr 06 2015
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, ...
%e 0, 1, 4, 9, 16, 25, 36, ...
%e 0, 1, 6, 19, 44, 85, 146, ...
%e 0, 1, 9, 39, 116, 275, 561, ...
%e 0, 1, 12, 69, 260, 751, 1812, ...
%e 0, 1, 16, 119, 560, 1955, 5552, ...
%p # First program:
%p h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
%p , t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
%p g:= proc(n, i, k, l)
%p `if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+
%p `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
%p end:
%p A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%p # second program:
%p gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
%p A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t (* First program: *)
%t h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten
%t (* second program: *)
%t gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 09 2013, translated from Maple *)
%Y Rows n=0-10 give: A000012, A001477, A000290, A005900, A139594, A210427, A210428, A210429, A210430, A210431, A210432.
%Y Columns k=0-8 give: A000007, A000012, A002620(n+2), A038163, A054498, A181477, A181478, A181479, A181480.
%Y Main diagonal gives: A209673.
%Y Cf. A138177, A191714.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Mar 20 2012
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