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A244297 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 3. 2
4, 10, 69, 195, 929, 3044, 11824, 40985, 158079, 539876, 2065087, 7272937, 27923757, 101194930, 381940222, 1429135919, 5607176733, 21323561733, 84260636527, 325309822037, 1337034045619, 5421586411034, 22509005469068, 92412147570641, 390023528935516 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,1

COMMENTS

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 3.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 5..150

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=3, `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$2, []):

seq(a(n), n=5..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, [[i]]}], {i, n}]];

g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 3, 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, 5, 35}] (* Jean-Fran├žois Alcover, Aug 28 2021, after Maple code *)

CROSSREFS

Column k=3 of A238707.

Sequence in context: A355375 A197939 A207160 * A207159 A152397 A239502

Adjacent sequences:  A244294 A244295 A244296 * A244298 A244299 A244300

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.)