A244296 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 2.

3, 6, 35, 71, 295, 751, 2326, 6524, 22309, 55992, 190282, 577410, 1951421, 5414977, 19405654, 64615030, 238446543, 726141375, 2682369977, 9475513873, 41043824531, 138540753071, 524631248766, 1902172512592, 8404692901429, 35025078519164, 148160349275671

Offset

4,1

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 4..250

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=2, `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, n, []):
seq(a(n), n=4..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 == 2, 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, 4, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

Crossrefs

Column k=2 of A238707.

Sequence in context: A134748 A068904 A309774 * A143046 A246069 A228279

Adjacent sequences: A244293 A244294 A244295 * A244297 A244298 A244299

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Status

approved