A384892 - OEIS

A384892

Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

1, 1, 2, 4, 13, 54, 314, 2120, 16700, 148333, 1468512, 16019532, 190899736, 2467007774, 34361896102, 513137616840, 8178130784179, 138547156531410, 2486151753462260, 47106033220679060, 939765362754015750, 19690321886243848784, 432292066866187743954

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OFFSET

0,3

LINKS

Christian Sievers, Table of n, a(n) for n = 0..450

FORMULA

a(n) = Sum_{d|n} A000255(d-1). - Christian Sievers, Jun 22 2025

EXAMPLE

The permutation (1,2,5,6,3,4,7,8) has maximal runs ((1,2),(5,6),(3,4),(7,8)), with lengths (2,2,2,2), so is counted under a(8).

The a(0) = 1 through a(4) = 13 permutations:

() (1) (12) (123) (1234)

(21) (132) (1324)

(213) (1432)

(321) (2143)

(2413)

(2431)

(3142)

(3214)

(3241)

(3412)

(4132)

(4213)

(4321)

MATHEMATICA

Table[Length[Select[Permutations[Range[n]], SameQ@@Length/@Split[#, #2==#1+1&]&]], {n, 0, 10}]

PROG

(PARI) a(n)=if(n, sumdiv(n, d, sum(i=0, d-1, (-1)^i*(d-i)!*binomial(d-1, i))), 1) \\ Christian Sievers, Jun 22 2025

CROSSREFS

For subsets instead of permutations we have A243815, for anti-runs A384889.

For strict partitions and distinct lengths we have A384178, anti-runs A384880.

For integer partitions and distinct lengths we have A384884, anti-runs A384885.

For distinct lengths we have A384891, for anti-runs A384907.

For partitions we have A384904, strict A384886.

A010027 counts permutations by maximal anti-runs, for runs A123513.

A034839 counts subsets by number of maximal runs, for strict partitions A116674.

A098859 counts Wilf partitions (distinct multiplicities), complement A336866.

A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.

Cf. A000255, A044813, A047993, A242882, A325325, A329739, A351202, A384175, A384176, A384177.

Sequence in context: A030886 A030810 A346550 * A058135 A030968 A030862

Adjacent sequences: A384889 A384890 A384891 * A384893 A384894 A384895

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 19 2025

EXTENSIONS

a(11) and beyond from Christian Sievers, Jun 22 2025

STATUS

approved