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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..120

Wikipedia, Young tableau

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 *)

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).

Sequence in context: A014009 A274220 A299881 * A374313 A288110 A085386

Adjacent sequences: A218290 A218291 A218292 * A218294 A218295 A218296

Keyword

nonn

Author

Joerg Arndt, Oct 25 2012

Status

approved