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A244299
Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 5.
2
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
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
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 25 2014
STATUS
approved