A244295 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 1.

2, 3, 14, 14, 69, 97, 251, 671, 1847, 2111, 12869, 33461, 58343, 189045, 841125, 2207347, 6651215, 12781755, 73096191, 308508927, 904926489, 1727792245, 7638794959, 44017642189, 177969495449, 522668483639, 1662245807549, 4496811662189, 32142974215379

Offset: 3

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

• nonn

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 1.

Examples

			a(4) = 3:
[1 2]   [1 3]   [1 4]
[3]     [2]     [2]
[4]     [4]     [3]
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..400
• Wikipedia, Young tableau

Crossrefs

Column k=1 of A238707.

Programs

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=1, `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=3..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 == 1, 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, 3, 35}] (* Jean-François Alcover, Aug 25 2021, after Maple code *)