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A244295
Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 1.
2
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
3,1
COMMENTS
Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 1.
LINKS
Wikipedia, Young tableau
EXAMPLE
a(4) = 3:
[1 2] [1 3] [1 4]
[3] [2] [2]
[4] [4] [3]
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 *)
CROSSREFS
Column k=1 of A238707.
Sequence in context: A287912 A206578 A056435 * A032806 A353870 A225756
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 25 2014
STATUS
approved