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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

R. Stanley, Parking Functions, 2011

Example

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.

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 *)

Crossrefs

Cf. A000041, A255047, A265016.

Sequence in context: A208715 A302408 A076700 * A026533 A131630 A305652

Adjacent sequences: A265004 A265005 A265006 * A265008 A265009 A265010

Keyword

nonn

Author

Alois P. Heinz, Nov 29 2015

Status

approved