A225121 Number of standard Young tableaux with shapes corresponding to partitions into distinct parts with minimal difference 2.

1, 1, 1, 1, 4, 5, 15, 21, 56, 246, 525, 1573, 5764, 14092, 41405, 136995, 772552, 2148290, 8806629, 31679365, 155743665, 495240074, 2049655762, 7403470138, 32627363920, 207316068370, 784695179515, 3721285661481, 16967347935561, 82192321793926, 455572563875425

Offset: 0

Author

Joerg Arndt, Apr 29 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..120
• Wikipedia, Young tableau

Crossrefs

Cf. A218293 (tableaux with shapes corresponding to partitions into distinct parts).

Cf. A000085 (standard Young tableaux for all shapes).

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) local s; s:=ceil(i*(i+2)/4);
      `if`(n=s, h([l[], seq(i-2*j, j=0..iquo(i-1,2))]), `if`(n>s, 0,
       g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-2, [l[], i]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..35);  # Alois P. Heinz, Apr 29 2013

Mathematica

h[l_List] := Module[{n}, n = Length[l]; Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_List] := Module[{s}, s = Ceiling[i*(i+2)/4]; If[n==s, h[Join[l, Table[i-2*j, {j, 0, Quotient[i-1, 2]}]]], If[n>s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-2, Append[l, i]]]]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jul 02 2015, after  Alois P. Heinz *)