A136662 Number of cycles of the permutations of [1,2,...,n].

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, 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, 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

Offset: 1

Author

Wolfdieter Lang, Feb 22 2008, May 21 2008

Keywords

• nonn
• easy
• tabf

Comments

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

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).

Examples

			Triangle begins:
  [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];
  ...
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.
		

Links

• Alois P. Heinz, Rows n = 1..8, flattened
• FindStat - Combinatorial Statistic Finder, The number of cycles in the cycle decomposition of a permutation
• Wolfdieter Lang, First rows and cycle decompositions.

Crossrefs

Row sums (total cycle numbers) A000254.

Cf. A130534.

Formula

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