A244298 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 4.

5, 15, 119, 421, 2254, 8999, 40349, 166817, 737829, 3008774, 13186593, 54944783, 238422808, 1010671048, 4395831546, 18821162274, 82799233661, 359711480525, 1599420076729, 7030074945271, 31626819884986, 141486845119777, 646988113794544, 2940338763342920

Offset

6,1

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 4.

Links

Alois P. Heinz, Table of n, a(n) for n = 6..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=4, `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=6..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 == 4, 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, 6, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

Crossrefs

Column k=4 of A238707.

Sequence in context: A220825 A303230 A173736 * A050542 A091096 A116957

Adjacent sequences: A244295 A244296 A244297 * A244299 A244300 A244301

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Status

approved