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A211362 Inversion sets of finite permutations interpreted as binary numbers. 4
0, 1, 4, 3, 6, 7, 32, 33, 20, 11, 22, 15, 48, 41, 52, 43, 30, 31, 56, 57, 60, 59, 62, 63, 512, 513, 516, 515, 518, 519, 288, 289, 148, 75, 150, 79, 304, 297, 180, 107, 158, 95, 312, 313, 188, 123, 190, 127, 768, 769, 644, 579, 646, 583, 800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Tilman Piesk, Table of n, a(n) for n = 0..5039

Wikipedia, Inversion (discrete mathematics)

EXAMPLE

The 4th finite permutation (2,3,1,4,...) has the inversion set {(1,3),(2,3)}. This set represented by a vector is (0,1,1,zeros...). This vector interpreted as a number is 6. So a(4)=6.

The 23rd finite permutation (4,3,2,1,...) has the inversion set {(1,2),(1,3),(2,3),(1,4),(2,4),(3,4)}. This set represented by a vector is (1,1,1,1,1,1,zeros...). This vector interpreted as a number is 63. So a(23)=63.

Beginning of corresponding array:

n    permutation   inversion set    a(n)

00     1 2 3 4     0  0 0  0 0 0     0

01     2 1 3 4     1  0 0  0 0 0     1

02     1 3 2 4     0  0 1  0 0 0     4

03     3 1 2 4     1  1 0  0 0 0     3

04     2 3 1 4     0  1 1  0 0 0     6

05     3 2 1 4     1  1 1  0 0 0     7

06     1 2 4 3     0  0 0  0 0 1    32

07     2 1 4 3     1  0 0  0 0 1    33

08     1 4 2 3     0  0 1  0 1 0    20

09     4 1 2 3     1  1 0  1 0 0    11

10     2 4 1 3     0  1 1  0 1 0    22

11     4 2 1 3     1  1 1  1 0 0    15

12     1 3 4 2     0  0 0  0 1 1    48

13     3 1 4 2     1  0 0  1 0 1    41

14     1 4 3 2     0  0 1  0 1 1    52

15     4 1 3 2     1  1 0  1 0 1    43

16     3 4 1 2     0  1 1  1 1 0    30

17     4 3 1 2     1  1 1  1 1 0    31

18     2 3 4 1     0  0 0  1 1 1    56

19     3 2 4 1     1  0 0  1 1 1    57

20     2 4 3 1     0  0 1  1 1 1    60

21     4 2 3 1     1  1 0  1 1 1    59

22     3 4 2 1     0  1 1  1 1 1    62

23     4 3 2 1     1  1 1  1 1 1    63

CROSSREFS

Cf. A018900, A055089, A195663, A034968, A211363.

Sequence in context: A071901 A103476 A021700 * A211246 A197829 A187890

Adjacent sequences:  A211359 A211360 A211361 * A211363 A211364 A211365

KEYWORD

nonn

AUTHOR

Tilman Piesk, Jun 03 2012

STATUS

approved

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Last modified April 24 03:45 EDT 2014. Contains 240947 sequences.