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A332852 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by decreasing lengths. 3
1, 7, 46, 341, 2871, 26797, 275353, 3090544, 37652660, 495756306, 7015094802, 106125820737, 1710625964061, 29267936828691, 529655709670675, 10110999740354242, 203072647138681534, 4280118000323963708, 94470690960204259548, 2179212745888578818307 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 3..450

Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.

Wikipedia, Permutation

FORMULA

a(n) = Sum_{k=0..floor(n/3)} k * A350015(n,k). - Alois P. Heinz, Dec 12 2021

MAPLE

b:= proc(n, l) option remember; `if`(n=0, l[3], add((j-1)!*b(n-j,

      sort([l[], j], `>`)[1..3])*binomial(n-1, j-1), j=1..n))

    end:

a:= n-> b(n, [0$3]):

seq(a(n), n=3..23);

MATHEMATICA

b[n_, l_] := b[n, l] = If[n == 0, l[[3]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 3]]] Binomial[n - 1, j - 1], {j, 1, n}]];

a[n_] := b[n, {0, 0, 0}];

a /@ Range[3, 23] (* Jean-Fran├žois Alcover, Mar 01 2020, after Alois P. Heinz *)

CROSSREFS

Column k=3 of A322384.

Cf. A350015.

Sequence in context: A240722 A067318 A072948 * A356827 A178962 A203267

Adjacent sequences:  A332849 A332850 A332851 * A332853 A332854 A332855

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 26 2020

STATUS

approved

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Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)