A265007 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n.

1, 1, 3, 7, 18, 40, 97, 216, 499, 1112, 2502, 5503, 12197, 26582, 58088, 125619, 271713, 583228, 1251115, 2668651, 5685053, 12059993, 25544291, 53926003, 113666195, 238946232, 501546514, 1050430420, 2196869731, 4586021745, 9560876381, 19900839742, 41373446190

Offset: 0

Author

Alois P. Heinz, Nov 29 2015

Keywords

• nonn

Examples

			The number of lambda-parking functions induced by the partitions of 4:
1 by [1,1,1,1]: [1,1,1,1],
4 by [1,1,2]: [1,1,1], [1,1,2], [1,2,1], [2,1,1],
4 by [2,2]: [1,1], [1,2], [2,1], [2,2],
5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
4 by [4]: [1], [2], [3], [4].
a(4) = 1 + 4 + 4 + 5 + 4 = 18.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50
• R. Stanley, Parking Functions, 2011

Crossrefs

Cf. A000041, A255047, A265016.

Programs

Maple

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, p([1$n, l[]]), g(n, i-1, l)
               +`if`(i>n, 0, g(n-i, i, [i, l[]]))):
a:= n-> g(n$2, []):
seq(a(n), n=0..20);

Mathematica

p[l_] := With[{n = Length[l]}, n! Det[Table[With[{t = j - i + 1},
     If[t < 0, 0, l[[i]]^t/t!]], {i, n}, {j, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, p[Join[
     Table[1, {n}], l]], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i, Prepend[l, i]]]];
a[n_] := If[n == 0, 1, g[n, n, {}]];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Aug 22 2021, after Alois P. Heinz *)