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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).
4
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 02 2015
STATUS
approved