A213242 Number of generalizations of the partition 1^n.

1, 2, 3, 6, 12, 31, 89, 303, 1119, 4649, 20572, 99241, 502622, 2725840, 15424019, 92211327, 571446565, 3716191974, 24920512847, 174169990243, 1251875604302, 9326245177768, 71241318920624, 562221733320241, 4535497053407716, 37677863148632647, 319551379756283637

Offset: 1

Author

Alois P. Heinz, Jun 14 2012

Keywords

• nonn

Comments

Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at any node in the poset and end at 1^n.

Examples

			For n=5 there are a(5) = 12 paths to 1^5 = 11111: 11111; 2111->11111; 221->2111->11111; 311->2111->11111; 32->221->2111->11111; 32->311->2111->11111; 41->221->2111->11111; 41->311->2111->11111; 5->32->221->2111->11111; 5->32->311->2111->11111; 5->41->221->2111->11111; 5->41->311->2111->11111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• Olivier Gérard, The ranked posets L(2),...,L(8)

Crossrefs

Cf. A002846, A213385, A213427, A213597.

Programs

Maple

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(n<2, 1, `if`(l[n]=0, b(subsop(n=NULL, l)),
      add(`if`(l[i]=0, 0, add(b([seq(l[t]-`if`(t=1, l[t],
      `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
      -1, 0)))), t=1..n)]), j=1..i/2)), i=2..n)))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);

Mathematica

b[l_] := b[l] = With[{n = Length[l]}, If[n < 2, 1, If[l[[n]] == 0, b[ReplacePart[l, n -> Sequence[] ]], Sum[If[l[[i]] == 0, 0, Sum[b[Join[Table[l[[t]]-If[t == 1, l[[t]], If[t == i, 1, If[t == j && t == i-j, -2, If[t == j || t == i-j, -1, 0]]]], {t, 1, n}]]], {j, 1, i/2}]], {i, 2, n}]]] ]; g[n_, i_, l_] := If[n == 0 && i == 0, b[l], If[i == 1, b[Prepend[l, n]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Prepend[ l, j]]], {j, 0, n/i}]]] ; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 27}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Extensions

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016