A218293 Number of standard Young tableaux with shapes corresponding to partitions into distinct parts.

1, 1, 1, 3, 4, 10, 31, 70, 190, 561, 2191, 6226, 22683, 74152, 283349, 1211354, 4572672, 18844177, 77585825, 327472752, 1418056071, 7083303437, 31251988918, 153456264178, 723293387594, 3596567095155, 17360616601051, 89955643932801, 486526881887485, 2551613423040841, 14029592127656040, 76756835252971657, 428044848852530252

Offset: 0

Author

Joerg Arndt, Oct 25 2012

Keywords

• nonn

Links

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

Crossrefs

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

Diagonal of A219272, row sums of A219274, A219311. - Alois P. Heinz, Nov 17 2012

Cf. A225121 (tableaux with shapes corresponding to partitions into distinct parts with minimal difference 2).

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:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 08 2012

Mathematica

h[l_List] := Module[{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=i*(i+1)/2}, If[n == s, h[Join[l, Table[i-j, {j, 0, i-1}]]], If[n > s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Append[l, i]]]]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)