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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 (list; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Table of n, a(n) for n = 3..400

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-Fran├žois Alcover, Aug 25 2021, after Maple code *)

CROSSREFS

Column k=1 of A238707.

Sequence in context: A287912 A206578 A056435 * A032806 A225756 A260143

Adjacent sequences:  A244292 A244293 A244294 * A244296 A244297 A244298

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Jun 25 2014

STATUS

approved

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Last modified January 22 12:12 EST 2022. Contains 350481 sequences. (Running on oeis4.)