A265145 Number of lambda-parking functions of the unique strict partition lambda with parts i_1<i_2<...<i_m and encoding n = Product_{j=1..m} prime(i_j-j+1).

1, 1, 2, 3, 3, 5, 4, 16, 8, 7, 5, 25, 6, 9, 12, 125, 7, 34, 8, 34, 16, 11, 9, 189, 15, 13, 50, 43, 10, 49, 11, 1296, 20, 15, 21, 243, 12, 17, 24, 253, 13, 64, 14, 52, 74, 19, 15, 1921, 24, 58, 28, 61, 16, 307, 27, 317, 32, 21, 17, 343, 18, 23, 98, 16807, 33

Offset

1,3

Comments

A strict partition is a partition into distinct parts.

Links

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

Richard P. Stanley, Parking Functions, 2011.

Example

n = 10 = 2*5 = prime(1)*prime(3) encodes strict partition [1,4] having seven lambda-parking functions: [1,1], [1,2], [2,1], [1,3], [3,1], [1,4], [4,1], thus a(10) = 7.

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)):
a:= n-> p((l-> [seq(l[j]+j-1, j=1..nops(l))])(sort([seq(
         numtheory[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[Function[l, Flatten[Table[l[[j]]+j-1,
     {j, 1, Length[l]}]]][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 *)

Crossrefs

Cf. A000009, A000040, A265144, A265146, A265208.

Sequence in context: A238792 A158745 A175108 * A103310 A046146 A081768

Adjacent sequences: A265142 A265143 A265144 * A265146 A265147 A265148

Keyword

nonn

Author

Alois P. Heinz, Dec 02 2015

Status

approved