cp's OEIS Frontend

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.)

Row n is the n-th finite permutation of {0,1,2,3...}.

Each finite permutation has a finite inversion set. The possible elements of the inversion sets are 2-element subsets of the integers, which can be ordered in an infinite sequence (compare A018900). Thus the inversion set can be represented by a binary vector, which can be interpreted as a binary number.

This sequence shows these numbers for the finite permutations in reverse colexicographic order (A055089, A195663). The corresponding inversion vectors are found in A007623. The corresponding inversion numbers (A034968) are the digit sums of the inversion vectors and the cardinality of the inversion sets, an thus also the binary digit sums of the numbers in this sequence.

This sequence is not monotonic. The permutation A211363 shows how the elements of this sequence (a) are ordered. a*A211363 gives the elements of a ordered by size.

Row n is the lexicographically earliest permutation of positive integers beginning with n. This also holds for the reverse colexicographic order, thus A007489(n-1) gives the position of n-th row of this array (which is one-based) in zero-based arrays A195663 & A055089.

The finite n X n square matrices in sequence A237265 converge towards this infinite square array.

Rows can be constructed also simply as follows: The first row is A000027 (natural numbers, also known as positive integers). For the n-th row, n=2, ..., pick n out from the terms of A000027 and move it to the front. This will create a permutation with one cycle of length n, in cycle notation: (1 n n-1 n-2 ... 3 2), which is the inverse of (1 2 ... n-1 n).

There are A000110(n) ways to choose n permutations from the n first rows of this table so that their composition is identity (counting all the different composition orders). This comment is essentially the same as my May 01 2006 comment on A000110, please see there for more information. - Antti Karttunen, Feb 10 2014

Also, for n > 1, the whole symmetric group S_n can be generated with just two rows, row 2, which is transposition (1 2), and row n, which is the inverse of cycle (1 ... n). See Rotman, p. 24, Exercise 2.9 (iii).

Equally, column 1 of A261097.

Take the n-th (n>=0) permutation from the list A055089 (A195663), change 1 to 2 and 2 to 1 to get another permutation, and note its rank in the same list to obtain a(n).

Self-inverse permutation of nonnegative integers.

The finite permutations whose position in reverse colexicographic order is A059590(n) (compare A055089, A195663) have the special feature that their inversion vectors (compare A007623) have only zeros and ones, and give 2*n when interpreted as binary numbers. As the inversion vectors are special, one may also take a look at the inversion sets. This sequence shows them, interpreted as binary numbers (compare A211362).