A179972 -id:A179972 - cp's OEIS frontend

A179973 Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).

1, 1, 2, 4, 12, 42, 216, 1200, 8664, 66384, 612264, 5910024, 66723384, 776642664, 10311400344, 141065450904, 2153769250584, 33743736435864, 583781959921944, 10308436641381144, 198863818304824344, 3914117125411211544, 83301822014343774744, 1805447764831655109144

Offset: 0

Alford Arnold, Aug 05 2010

nonn

The original name was: Row sums of A179972 and also of A179974.

			a(4) = 12 = 6 + 2 + 2 + 1 + 1: (1234), (1243), (1324), (1342), (1423), (1432),
  (13)(24), (14)(23), (1)(234), (1)(243), (1)(2)(34), (1)(2)(3)(4).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000041, A000142, A000522, A004086, A053529, A179972, A179974, A327711, A362362, A364128.

a:= n-> add((n-nops(p))!, p=combinat[partition](n)):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
     (p-n)!, b(n, i-1, p)+b(n-i, min(n-i, i), p-1))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p - n)!, b[n, i - 1, p] + b[n - i, Min[n - i, i], p - 1]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

From Alois P. Heinz, Jul 09 2023: (Start)
a(n) = Sum_{lambda in partitions(n)} (n - |lambda|)!.
Limit_{n->oo} A004086(a(n))/10^A055642(a(n)) = A364128. (End)

Edited by R. J. Mathar, May 17 2016

a(0), a(9)-a(23) and new name from Alois P. Heinz, Jul 09 2023

A179974 Triangle read by rows: T(n,k) = (n-A049085(n,k))! in columns 1<=k<=A000041(n), rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 6, 1, 1, 2, 2, 6, 6, 24, 1, 1, 2, 6, 2, 6, 24, 6, 24, 24, 120, 1, 1, 2, 6, 2, 6, 24, 24, 6, 24, 120, 24, 120, 120, 720, 1, 1, 2, 6, 24, 2, 6, 24, 24, 120, 6, 24, 120, 120, 720, 24, 120, 720, 120, 720, 720, 5040, 1, 1, 2, 6, 24, 2, 6, 24, 120, 24, 120, 720, 6, 24, 120, 120, 720, 720, 24, 120, 720, 720, 5040, 120, 720, 5040

Offset: 1

Alford Arnold, Aug 05 2010

Since A049085 is a resortment of A036043 both A179972 and A179974 have row sums equal to A179973.

			Triangle begins
1;
1,1;
1,1,2;
1,1,2,2,6;
1,1,2,2,6,6,24;
1,1,2,6,2,6,24,6,24,24,120;
1,1,2,6,2,6,24,24,6,24,120,24,120,120,720;
1,1,2,6,24,2,6,24,24,120,6,24,120,120,720,24,120,720,120,720,720,5040;
1,1,2,6,24,2,6,24,120,24,120,720,6,24,120,120,720,720,24,120,720,720,5040,120,720,5040,720,5040,5040,40320,

Cf. A000041 (row lengths), A179973 (row sums), A036042, A049085 (max part).

cp's OEIS Frontend

A179973 Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions