A210391 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.

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, 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, 64, 231, 561, 751, 560, 189, 25, 1, 0

Offset: 0

Author

Alois P. Heinz, Mar 20 2012

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   4,   9,  16,   25,   36, ...
  0,  1,   6,  19,  44,   85,  146, ...
  0,  1,   9,  39, 116,  275,  561, ...
  0,  1,  12,  69, 260,  751, 1812, ...
  0,  1,  16, 119, 560, 1955, 5552, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
• Wikipedia, Young tableau

Crossrefs

Rows n=0-10 give: A000012, A001477, A000290, A005900, A139594, A210427, A210428, A210429, A210430, A210431, A210432.

Columns k=0-8 give: A000007, A000012, A002620(n+2), A038163, A054498, A181477, A181478, A181479, A181480.

Main diagonal gives: A209673.

Cf. A138177, A191714.

Programs

Maple

# First program:
h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
                 , t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
g:= proc(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, [l[], i]))))
    end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second program:
gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

(* First program: *)
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
(* second program: *)
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 *)

Formula

G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).

A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - Alois P. Heinz, Apr 06 2015