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
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
Programs
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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
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