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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. 21
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Wikipedia, Young tableau

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Dec 09 2013, translated from Maple *)

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.

Sequence in context: A323224 A118340 A213276 * A071921 A003992 A246118

Adjacent sequences:  A210388 A210389 A210390 * A210392 A210393 A210394

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 20 2012

STATUS

approved

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Last modified May 28 17:28 EDT 2020. Contains 334684 sequences. (Running on oeis4.)