A265144 - OEIS

A265144

Number of lambda-parking functions of the unique partition lambda with encoding n = Product_{i:lambda} prime(i).

1, 1, 2, 1, 3, 3, 4, 1, 4, 5, 5, 4, 6, 7, 8, 1, 7, 7, 8, 7, 12, 9, 9, 5, 9, 11, 8, 10, 10, 16, 11, 1, 16, 13, 15, 11, 12, 15, 20, 9, 13, 25, 14, 13, 20, 17, 15, 6, 16, 19, 24, 16, 16, 15, 21, 13, 28, 19, 17, 27, 18, 21, 32, 1, 27, 34, 19, 19, 32, 34, 20, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Richard P. Stanley, Parking Functions, 2011.

EXAMPLE

n = 18 = 2*3*3 = prime(1)*prime(2)*prime(2) encodes partition [1,2,2] having seven lambda-parking functions: [1,1,1], [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], thus a(18) = 7.

MAPLE

with(numtheory):

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)

-> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):

a:= n-> p(sort([seq(pi(i[1])$i[2], i=ifactors(n)[2])])):

seq(a(n), n=1..100);

MATHEMATICA

p[l_] := Function[n, n! Det[Table[Function[t, If[t<0, 0,

l[[i]]^t/t!]][j-i+1], {i, n}, {j, n}]]][Length[l]];

a[n_] := If[n==1, 1, p[Sort[Flatten[Table[Table[

PrimePi[i[[1]]], {i[[2]]}], {i, FactorInteger[n]}]]]]];

Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 21 2021, after Alois P. Heinz *)