A244305 Number of standard Young tableaux with 2n cells such that the lengths of the first and the last row differ by n.

1, 0, 3, 10, 119, 791, 8823, 87515, 1042823, 12448912, 166443706, 2246438833, 32782857721, 488717384754, 7695520330054, 124248088106249, 2091672883631855, 36107381616662300, 644987804706582806, 11799406380611542654, 222235188242044718908, 4280160250751484220674

Offset

0,3

Comments

Also the number of ballot sequences of length 2n such that the multiplicities of the largest and the smallest value differ by n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

Formula

a(n) = A238707(2n,n).

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, k, l) `if`(n=0 or i<1 or `if`(l<>[], l[1], i)-1<k, 0,
      `if`(l<>[] and l[1]-i=k, `if`(irem(n, i, 'j')=0, h([l[], i$j]),
       0), add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 1, g(2*n$2, n, [])):
seq(a(n), n=0..25);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+
     Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, k_, l_] := If[n == 0 || i<1 || If[l != {}, l[[1]], i]-1<k, 0,
     If[l != {} && l[[1]] - i == k, j = Quotient[n, i];
     If[Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0],
     Sum[g[n - i*j, i-1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 1, g[2n, 2n, n, {}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 29 2021, after Maple code *)

Crossrefs

Cf. A238707.

Sequence in context: A132480 A201694 A336210 * A074260 A180978 A241461

Adjacent sequences: A244302 A244303 A244304 * A244306 A244307 A244308

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Status

approved