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A244305
Number of standard Young tableaux with 2n cells such that the lengths of the first and the last row differ by n.
2
1, 0, 3, 10, 119, 791, 8823, 87515, 1042823, 12448912, 166443706, 2246438833, 32782857721, 488717384754, 7695520330054, 124248088106249, 2091672883631855, 36107381616662300, 644987804706582806, 11799406380611542654, 222235188242044718908, 4280160250751484220674
OFFSET
0,3
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
FORMULA
a(n) = A238707(2n,n).
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<k, 0,
`if`(l<>[] 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]-1<k, 0,
If[l != {} && l[[1]] - i == k, j = Quotient[n, i];
If[Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0],
Sum[g[n - i*j, i-1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 1, g[2n, 2n, n, {}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 29 2021, after Maple code *)
CROSSREFS
Cf. A238707.
Sequence in context: A132480 A201694 A336210 * A074260 A180978 A241461
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 25 2014
STATUS
approved