A265202 Total number of lambda-parking functions induced by all partitions of n into distinct parts.

1, 1, 2, 6, 9, 15, 36, 53, 78, 119, 286, 401, 591, 829, 1232, 2910, 4084, 5789, 8070, 11281, 15823, 37747, 51622, 72919, 98986, 136600, 181648, 254638, 586891, 799841, 1110303, 1495279, 2018749, 2657612, 3552560, 4738775, 10857521, 14560375, 20061359, 26603227

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

R. Stanley, Parking Functions, 2011

Example

a(0) = 1: [].
a(1) = 1: [1].
a(2) = 2: [1], [2].
a(3) = 6: [1], [2], [3], [1,1], [1,2], [2,1].
a(4) = 9: [1], [2], [3], [4], [1,1], [1,2], [1,3], [2,1], [3,1].
a(5) = 15: [1], [2], [3], [4], [5], [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1].
a(6) = 36: [1], [2], [3], [4], [5], [6], [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1], [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].

Maple

b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!)+
      `if`(n<t, 0, add(b(p+1, `if`(i=j, g+1, 1), n-max(j, t), j,
       max(j+1, t+1))/`if`(i=j, 1, g!), j=i..n))
    end:
a:= n-> `if`(n=0, 1, b(0$2, n, 1$2)):
seq(a(n), n=0..50);

Mathematica

b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!] + If[n<t, 0, Sum[b[p+1, If[i==j, g+1, 1], n-Max[j, t], j, Max[j+1, t+1]]/If[i==j, 1, g!], {j, i, n}]]; a[n_] := If[n==0, 1, b[0, 0, n, 1, 1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Crossrefs

Row sums of A265208.

Cf. A000009, A255047, A265016.

Sequence in context: A103139 A181025 A345051 * A355738 A329743 A320496

Adjacent sequences: A265199 A265200 A265201 * A265203 A265204 A265205

Keyword

nonn

Author

Alois P. Heinz, Dec 04 2015

Status

approved