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A244296 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 2. 2
3, 6, 35, 71, 295, 751, 2326, 6524, 22309, 55992, 190282, 577410, 1951421, 5414977, 19405654, 64615030, 238446543, 726141375, 2682369977, 9475513873, 41043824531, 138540753071, 524631248766, 1902172512592, 8404692901429, 35025078519164, 148160349275671 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,1

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 4..250

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=2, `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=4..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 == 2, 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, 4, 35}] (* Jean-Fran├žois Alcover, Aug 28 2021, after Maple code *)

CROSSREFS

Column k=2 of A238707.

Sequence in context: A134748 A068904 A309774 * A143046 A246069 A228279

Adjacent sequences:  A244293 A244294 A244295 * A244297 A244298 A244299

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Jun 25 2014

STATUS

approved

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Last modified August 14 23:38 EDT 2022. Contains 356122 sequences. (Running on oeis4.)