login
Number of cycles of the permutations of [1,2,...,n].
3

%I #24 Jan 26 2023 14:33:51

%S 1,2,1,3,2,2,1,1,2,4,3,3,2,2,3,3,2,2,1,1,2,2,1,3,2,2,1,1,2,2,3,1,2,5,

%T 4,4,3,3,4,4,3,3,2,2,3,3,2,4,3,3,2,2,3,3,4,2,3,4,3,3,2,2,3,3,2,2,1,1,

%U 2,2,1,3,2,2,1,1,2,2,3,1,2,3,2,2,1,1,2,4,3,3,2,2,3,3,2,2,1,1,2,2,3,1,2,2,1

%N Number of cycles of the permutations of [1,2,...,n].

%C The row lengths sequence is A000142(n), n>=1, (factorials).

%C The permutations of [1,2,...,n] are ordered in the standard way (lexicographic or numerically increasing). E.g., in Maple as permute(n) list for not too large n (around 10).

%H Alois P. Heinz, <a href="/A136662/b136662.txt">Rows n = 1..8, flattened</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000031">The number of cycles in the cycle decomposition of a permutation</a>

%H Wolfdieter Lang, <a href="/A136662/a136662.txt"> First rows and cycle decompositions</a>.

%F a(n,k) = number of cycles of the k-th permutation of [1,2,...,n] in standard (increasing) order.

%e Triangle begins:

%e [1];

%e [2,1];

%e [3,2,2,1,1,2];

%e [4,3,3,2,2,3,3,2,2,1,1,2,2,1,3,2,2,1,1,2,2,3,1,2];

%e ...

%e Row n=3: permutations of [1,2,3] in the order [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]. Cycle decomposition: [[[1], [2], [3]], [[1], [2, 3]], [[1, 2], [3]], [[1, 2, 3]], [[1, 3, 2]], [[1, 3], [2]]]. Number of cycles: [3,2,2,1,1,2], the entries of row n=3.

%Y Row sums (total cycle numbers) A000254.

%Y Cf. A130534.

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Feb 22 2008, May 21 2008