A244299 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 5.

6, 21, 188, 791, 4696, 21614, 109745, 513421, 2557358, 11885545, 58291639, 275421640, 1342532532, 6411950652, 31310737486, 151220406569, 742729520457, 3625802212441, 17956348335989, 88575381634494, 442565032597013, 2207206278880826, 11138577085071310

Offset

7,1

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 5.

Links

Alois P. Heinz, Table of n, a(n) for n = 7..100

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 j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=5, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=7..35);

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_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 5, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 7, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

Crossrefs

Column k=5 of A238707.

Sequence in context: A179768 A131960 A372137 * A143049 A213680 A164643

Adjacent sequences: A244296 A244297 A244298 * A244300 A244301 A244302

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Status

approved