A244305 Number of standard Young tableaux with 2n cells such that the lengths of the first and the last row differ by n.

1, 0, 3, 10, 119, 791, 8823, 87515, 1042823, 12448912, 166443706, 2246438833, 32782857721, 488717384754, 7695520330054, 124248088106249, 2091672883631855, 36107381616662300, 644987804706582806, 11799406380611542654, 222235188242044718908, 4280160250751484220674

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60

Crossrefs

Cf. 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, k, l) `if`(n=0 or i<1 or `if`(l<>[], l[1], i)-1[] and l[1]-i=k, `if`(irem(n, i, 'j')=0, h([l[], i$j]),
       0), add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 1, g(2*n$2, n, [])):
seq(a(n), n=0..25);

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_, k_, l_] := If[n == 0 || i<1 || If[l != {}, l[[1]], i]-1Jean-François Alcover, Aug 29 2021, after Maple code *)

Formula

a(n) = A238707(2n,n).