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A181897 Triangle of refined rencontres numbers: T(n,k) is the number of permutations of n elements with cycle type k (k-th integer partition, defined by A194602). 4
1, 1, 1, 1, 3, 2, 1, 6, 8, 3, 6, 1, 10, 20, 15, 30, 20, 24, 1, 15, 40, 45, 90, 120, 144, 15, 90, 40, 120, 1, 21, 70, 105, 210, 420, 504, 105, 630, 280, 840, 210, 504, 420, 720, 1, 28, 112, 210, 420, 1120, 1344, 420, 2520, 1120, 3360, 1680, 4032 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

T(n,k) tells how often k appears among the first n! entries of A198380, i.e., how many permutations of n elements have the cycle type denoted by k.

This triangle is a refinement of the rencontres numbers A008290, which tell only how many permutations of n elements actually move a certain number of elements. How many of these permutations have a certain cycle type is a more detailed question, answered by this triangle.

The rows are counted from 1, the columns from 0.

Row lengths: 1, 2, 3, 5, 7, 11, ... (partition numbers A000041).

Row sums: 1, 2, 6, 24, 120, 720, ... (factorial numbers A000142).

Row maxima: 1, 1, 3, 8, 30, 144, ... (A059171).

Distinct entries per row: 1, 1, 3, 4, 6, 7, ... (A073906).

LINKS

Table of n, a(n) for n=1..57.

Tilman Piesk, Permutations by cycle type (Wikiversity article)

FORMULA

T(n,1) = A000217(n).

T(n,2) = A007290(n).

Let m2, m3, ... count the appearances of 2, 3, ... in the cycle type. E.g., the cycle type 2, 2, 2, 3, 3, 4 implies m2=3, m3=2, m4=1. Then T(n;m2,m3,m4,...) = n!/((2^m2 3^m3 4^m4 ...) m1!m2!m3!m4! ...) where m1 = n - 2m2 - 3m3 - 4m4 - ... . - Carlos Mafra, Nov 25 2014

EXAMPLE

Triangle begins:

1;

1,  1;

1,  3,  2;

1,  6,  8,  3,  6;

1, 10, 20, 15, 30,  20,  24;

1, 15, 40, 45, 90, 120, 144, 15, 90, 40, 120

MATHEMATICA

Table[CoefficientRules[ n! CycleIndex[SymmetricGroup[n], s] // Expand][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Nov 09 2014 *)

CROSSREFS

Cf. A000041, A000142, A198380, A194602, A008290, A073906, A000217, A007290.

Sequence in context: A114586 A052174 A227790 * A212207 A111049 A211955

Adjacent sequences:  A181894 A181895 A181896 * A181898 A181899 A181900

KEYWORD

tabf,nonn

AUTHOR

Tilman Piesk, Mar 31 2012

STATUS

approved

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Last modified September 20 08:08 EDT 2019. Contains 327214 sequences. (Running on oeis4.)