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Identifier

Mp00329: Permutations —Tanimoto⟶ Permutations

Images

[1] => [1]

[1,2] => [1,2]

[2,1] => [2,1]

[1,2,3] => [1,2,3]

[1,3,2] => [2,1,3]

[2,1,3] => [1,3,2]

[2,3,1] => [3,1,2]

[3,1,2] => [2,3,1]

[3,2,1] => [3,2,1]

[1,2,3,4] => [1,2,3,4]

[1,2,4,3] => [2,3,1,4]

[1,3,2,4] => [1,2,4,3]

[1,3,4,2] => [2,4,1,3]

[1,4,2,3] => [2,1,3,4]

[1,4,3,2] => [2,1,4,3]

[2,1,3,4] => [1,3,2,4]

[2,1,4,3] => [3,2,1,4]

[2,3,1,4] => [1,3,4,2]

[2,3,4,1] => [3,4,1,2]

[2,4,1,3] => [3,1,2,4]

[2,4,3,1] => [3,1,4,2]

[3,1,2,4] => [1,4,2,3]

[3,1,4,2] => [4,2,1,3]

[3,2,1,4] => [1,4,3,2]

[3,2,4,1] => [4,3,1,2]

[3,4,1,2] => [4,1,2,3]

[3,4,2,1] => [4,1,3,2]

[4,1,2,3] => [2,3,4,1]

[4,1,3,2] => [2,4,3,1]

[4,2,1,3] => [3,2,4,1]

[4,2,3,1] => [3,4,2,1]

[4,3,1,2] => [4,2,3,1]

[4,3,2,1] => [4,3,2,1]

[1,2,3,4,5] => [1,2,3,4,5]

[1,2,3,5,4] => [2,3,4,1,5]

[1,2,4,3,5] => [1,2,3,5,4]

[1,2,4,5,3] => [2,3,5,1,4]

[1,2,5,3,4] => [2,3,1,4,5]

[1,2,5,4,3] => [2,3,1,5,4]

[1,3,2,4,5] => [1,2,4,3,5]

[1,3,2,5,4] => [2,4,3,1,5]

[1,3,4,2,5] => [1,2,4,5,3]

[1,3,4,5,2] => [2,4,5,1,3]

[1,3,5,2,4] => [2,4,1,3,5]

[1,3,5,4,2] => [2,4,1,5,3]

[1,4,2,3,5] => [1,2,5,3,4]

[1,4,2,5,3] => [2,5,3,1,4]

[1,4,3,2,5] => [1,2,5,4,3]

[1,4,3,5,2] => [2,5,4,1,3]

[1,4,5,2,3] => [2,5,1,3,4]

[1,4,5,3,2] => [2,5,1,4,3]

[1,5,2,3,4] => [2,1,3,4,5]

[1,5,2,4,3] => [2,1,3,5,4]

[1,5,3,2,4] => [2,1,4,3,5]

[1,5,3,4,2] => [2,1,4,5,3]

[1,5,4,2,3] => [2,1,5,3,4]

[1,5,4,3,2] => [2,1,5,4,3]

[2,1,3,4,5] => [1,3,2,4,5]

[2,1,3,5,4] => [3,2,4,1,5]

[2,1,4,3,5] => [1,3,2,5,4]

[2,1,4,5,3] => [3,2,5,1,4]

[2,1,5,3,4] => [3,2,1,4,5]

[2,1,5,4,3] => [3,2,1,5,4]

[2,3,1,4,5] => [1,3,4,2,5]

[2,3,1,5,4] => [3,4,2,1,5]

[2,3,4,1,5] => [1,3,4,5,2]

[2,3,4,5,1] => [3,4,5,1,2]

[2,3,5,1,4] => [3,4,1,2,5]

[2,3,5,4,1] => [3,4,1,5,2]

[2,4,1,3,5] => [1,3,5,2,4]

[2,4,1,5,3] => [3,5,2,1,4]

[2,4,3,1,5] => [1,3,5,4,2]

[2,4,3,5,1] => [3,5,4,1,2]

[2,4,5,1,3] => [3,5,1,2,4]

[2,4,5,3,1] => [3,5,1,4,2]

[2,5,1,3,4] => [3,1,2,4,5]

[2,5,1,4,3] => [3,1,2,5,4]

[2,5,3,1,4] => [3,1,4,2,5]

[2,5,3,4,1] => [3,1,4,5,2]

[2,5,4,1,3] => [3,1,5,2,4]

[2,5,4,3,1] => [3,1,5,4,2]

[3,1,2,4,5] => [1,4,2,3,5]

[3,1,2,5,4] => [4,2,3,1,5]

[3,1,4,2,5] => [1,4,2,5,3]

[3,1,4,5,2] => [4,2,5,1,3]

[3,1,5,2,4] => [4,2,1,3,5]

[3,1,5,4,2] => [4,2,1,5,3]

[3,2,1,4,5] => [1,4,3,2,5]

[3,2,1,5,4] => [4,3,2,1,5]

[3,2,4,1,5] => [1,4,3,5,2]

[3,2,4,5,1] => [4,3,5,1,2]

[3,2,5,1,4] => [4,3,1,2,5]

[3,2,5,4,1] => [4,3,1,5,2]

[3,4,1,2,5] => [1,4,5,2,3]

[3,4,1,5,2] => [4,5,2,1,3]

[3,4,2,1,5] => [1,4,5,3,2]

[3,4,2,5,1] => [4,5,3,1,2]

[3,4,5,1,2] => [4,5,1,2,3]

[3,4,5,2,1] => [4,5,1,3,2]

[3,5,1,2,4] => [4,1,2,3,5]

[3,5,1,4,2] => [4,1,2,5,3]

>>> Load all 5913 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.

References

[1] Tanimoto, S. An operator on permutations and its application to Eulerian numbers MathSciNet:1829751 zbMATH:0986.05005 DOI:10.1006/eujc.2000.0497
[2] Tanimoto, S. A study of Eulerian numbers by means of an operator on permutations MathSciNet:1957962 zbMATH:1014.05007 DOI:10.1016/S0195-6698(02)00137-3
[3] Tanimoto, S. On the numbers of orbits of permutations under an operator related to Eulerian numbers MathSciNet:2079934 zbMATH:1048.05004 DOI:10.1007/s00026-004-0218-2
[4] Tanimoto, S. A New Approach to Signed Eulerian Numbers arXiv:math/0602263
[5] Tanimoto, S. A study of Eulerian numbers for permutations in the alternating group MathSciNet:2264846 zbMATH:1104.11011
[6] Tanimoto, S. Combinatorics of the group of parity alternating permutations MathSciNet:2593308 zbMATH:1218.05007 DOI:10.1016/j.aam.2009.07.002
[7] Tanimoto, S. Parity alternating permutations and signed Eulerian numbers MathSciNet:2737324 zbMATH:1233.05217 DOI:10.1007/s00026-010-0064-3

Properties

bijective

Sage code

def mapping(pi):
    n = len(pi)
    # If n is in the front then remove n, add 1 to each entry,
    # and place 1 at the back
    if pi[0] == n:
        return Permutation([i + 1 for i in pi[1:n]] + [1])
    # If n is at the back then remove n, add 1 to each entry,
    # and place 1 at the front
    elif pi[n-1] == n:
        return Permutation([1] + [i + 1 for i in pi[0:n-1]])
    # Otherwise add 1 to each entry, changing n to 1 instead of n+1
    else:
        return Permutation([(i % n) + 1 for i in pi])

Weight

Created

Mar 02, 2025 at 01:16 by Nathan R. Krause

Updated

Mar 06, 2025 at 20:00 by Nathan R. Krause