A198380 Cycle type of the n-th finite permutation represented by index number of A194602.

0, 1, 1, 2, 2, 1, 1, 3, 2, 4, 4, 2, 2, 4, 1, 2, 3, 4, 4, 2, 2, 1, 4, 3, 1, 3, 3, 5, 5, 3, 2, 5, 4, 6, 6, 4, 4, 6, 2, 4, 5, 6, 6, 4, 4, 2, 6, 5, 2, 5, 4, 6, 6, 4, 1, 3, 2, 4, 4, 2, 3, 5, 4, 6, 6, 5, 5, 3, 6, 4, 5, 6, 4, 6, 2, 4, 5, 6, 2, 4, 1, 2, 3, 4, 4, 6

Offset: 0

Author

Tilman Piesk, Oct 23 2011

Keywords

• nonn

Comments

This sequence shows the cycle type of each finite permutation (A195663) as the index number of the corresponding partition. (When a permutation has a 3-cycle and a 2-cycle, this corresponds to the partition 3+2, etc.) Partitions can be ordered, so each partition can be denoted by its index in this order, e.g. 6 for the partition 3+2. Compare A194602.

From the properties of A194602 follows:

Entries 1,2,4,6,10,14,21... ( A000041(n)-1 from n=2 ) correspond to permutations with exactly one n-cycle (and no other cycles).

Entries 1,3,7,15,30,56,101... ( A000041(2n-1) from n=1 ) correspond to permutations with exactly n 2-cycles (and no other cycles), so these are the symmetric permutations.

Entries n = 1,3,4,7,9,10,12... ( A194602(n) has an even binary digit sum ) correspond to even permutations. This goes along with the fact, that a permutation is even when its partition contains an even number of even addends.

(Compare "Table for A194602" in section LINKS. Concerning the first two properties see especially the end of this file.)

Links

• Tilman Piesk, Table of n, a(n) for n = 0..5039
• Tilman Piesk, Table including permutations of 8 elements and partitions written as sums for n = 0..40319
• Tilman Piesk, Permutations by cycle type (Wikiversity article)
• Tilman Piesk, Table for A194602

Crossrefs

Cf. A195663, A195664, A055089 (ordered finite permutations).

Cf. A194602 (ordered partitions interpreted as binary numbers).

Cf. A181897 (number of n-permutations with cycle type k).

Extensions

Changed offset to 0 by Tilman Piesk, Jan 25 2012