A244297 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 3.

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

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-François 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