Identifier
- St000124: Permutations ⟶ ℤ
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 0
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 6
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 0
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
[1,4,5,3,2] => 0
[1,5,2,3,4] => 6
[1,5,2,4,3] => 0
[1,5,3,2,4] => 0
[1,5,3,4,2] => 0
[1,5,4,2,3] => 0
[1,5,4,3,2] => 0
[2,1,3,4,5] => 1
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 1
[2,1,5,3,4] => 2
[2,1,5,4,3] => 0
[2,3,1,4,5] => 1
[2,3,1,5,4] => 1
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 2
[2,3,5,4,1] => 0
[2,4,1,3,5] => 2
[2,4,1,5,3] => 2
[2,4,3,1,5] => 0
[2,4,3,5,1] => 0
[2,4,5,1,3] => 2
[2,4,5,3,1] => 0
[2,5,1,3,4] => 6
[2,5,1,4,3] => 0
[2,5,3,1,4] => 0
[2,5,3,4,1] => 0
[2,5,4,1,3] => 0
[2,5,4,3,1] => 0
[3,1,2,4,5] => 2
[3,1,2,5,4] => 2
[3,1,4,2,5] => 2
[3,1,4,5,2] => 2
[3,1,5,2,4] => 4
[3,1,5,4,2] => 0
[3,2,1,4,5] => 0
[3,2,1,5,4] => 0
[3,2,4,1,5] => 0
[3,2,4,5,1] => 0
[3,2,5,1,4] => 0
[3,2,5,4,1] => 0
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 0
[3,4,2,5,1] => 0
[3,4,5,1,2] => 2
[3,4,5,2,1] => 0
[3,5,1,2,4] => 6
[3,5,1,4,2] => 0
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Description
The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map S that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of S−1(π) for each permutation π.
The map S can also be realized using the quotient of the 0-Hecke Monoid of the symmetric group by the relation πiπi+1πi=πi+1πi, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map S that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of S−1(π) for each permutation π.
The map S can also be realized using the quotient of the 0-Hecke Monoid of the symmetric group by the relation πiπi+1πi=πi+1πi, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
References
[1] Bóna, Miklós Combinatorics of permutations MathSciNet:2919720
Code
def statistic(x):
return sum(1 for pi in Permutations(x.size()) if pi.simion_schmidt([3,2,1]) == x)
Created
Jun 18, 2013 at 18:08 by Tom Denton
Updated
Mar 17, 2018 at 16:36 by Martin Rubey
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