A195664 -id:A195664 - cp's OEIS frontend

A195663 Array read by antidiagonals: Consecutive finite permutations of positive integers in reverse colexicographic order.

1, 2, 2, 3, 1, 1, 4, 3, 3, 3, 5, 4, 2, 1, 2, 6, 5, 4, 2, 3, 3, 7, 6, 5, 4, 1, 2, 1, 8, 7, 6, 5, 4, 1, 2, 2, 9, 8, 7, 6, 5, 4, 4, 1, 1, 10, 9, 8, 7, 6, 5, 3, 4, 4, 4, 11, 10, 9, 8, 7, 6, 5, 3, 2, 1, 2, 12, 11, 10, 9, 8, 7, 6, 5, 3, 2, 4, 4, 13, 12, 11, 10, 9, 8, 7, 6, 5, 3, 1, 2, 1, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 3, 1, 3, 3

Offset: 0

Tilman Piesk, Sep 22 2011

Row n is the n-th finite permutation of {1,2,3,4,...}.

			The first 24 permutations of positive integers in rev colex order:
00  -->  1 2 3 4 5 6 7 8 ...
01  -->  2 1 3 4 ...
02  -->  1 3 2 4 ...
03  -->  3 1 2 4 ...
04  -->  2 3 1 4 ...
05  -->  3 2 1 4 ...
06  -->  1 2 4 3 ...
07  -->  2 1 4 3 ...
08  -->  1 4 2 3 ...
09  -->  4 1 2 3 ...
10  -->  2 4 1 3 ...
11  -->  4 2 1 3 ...
12  -->  1 3 4 2 ...
13  -->  3 1 4 2 ...
14  -->  1 4 3 2 ...
15  -->  4 1 3 2 ...
16  -->  3 4 1 2 ...
17  -->  4 3 1 2 ...
18  -->  2 3 4 1 ...
19  -->  3 2 4 1 ...
20  -->  2 4 3 1 ...
21  -->  4 2 3 1 ...
22  -->  3 4 2 1 ...
23  -->  4 3 2 1 ...

Tilman Piesk, Table of n, a(n) for n = 0..7259
Tilman Piesk, Detailed table of the 24 permutations of 1...4
Tilman Piesk, Table of the 40320 permutations of 1...8, a supporting file of A198380
OEIS-Wiki, Orderings section rev colex
Tilman Piesk, MATLAB code used for the calculation

Cf. A055089 (a very compact representation of these permutations).

Cf. A195664 (same for nonnegative integers, so all entries are smaller by 1).

a(n) = A195664(n)+1.

A055090 Number of cycles (excluding fixed points) of the n-th finite permutation in reversed colexicographic ordering (A055089).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Antti Karttunen, Apr 18 2000

nonn

Among the first n! entries k appears A136394(n,k) times. - Tilman Piesk, Apr 06 2012

Tilman Piesk, Table of n, a(n) for n = 0..5039
Tilman Piesk, Table for A198380 (a[n] is the number of addends in parentheses on the right of this table)
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)
Index entries for sequences related to factorial base representation

Cf. A195663, A195664, A055089 (ordered finite permutations).
Cf. A198380 (cycle type of the n-th finite permutation).
Cf. A136394, A055089, A055090, A055093, A055091, A060128, A290095.

with(group); seq(nops(convert(PermRevLexUnrank(j),'disjcyc')),j=0..)];
# Procedure PermRevLexUnrank given in A055089.

a(n) = A055093(n) - A055091(n).

a(n) = A056170(A290095(n)) = A060128(A060126(n)). - Antti Karttunen, Dec 30 2017

Name changed by Tilman Piesk, Apr 06 2012

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

Original entry on oeis.org

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

Tilman Piesk, Oct 23 2011

nonn

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.)

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

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).

Changed offset to 0 by Tilman Piesk, Jan 25 2012

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A195663 Array read by antidiagonals: Consecutive finite permutations of positive integers in reverse colexicographic order.

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A055090 Number of cycles (excluding fixed points) of the n-th finite permutation in reversed colexicographic ordering (A055089).

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

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