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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000124
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(load all 4 compositions to match this statistic)
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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 $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
Matching statistic: St000205
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? = 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 1
[1,2,3] => [3]
=> []
=> ?
=> ? = 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 1
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 1
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 1
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 6
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? = 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 2
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 6
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? = 2
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? = 6
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? = 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 1
[1,2,3] => [3]
=> []
=> ?
=> ? = 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 1
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 1
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 1
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 6
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? = 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 1
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 1
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 2
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 6
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? = 1
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? = 2
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? = 2
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? = 6
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000781
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Values
[1] => [1]
=> []
=> ?
=> ? = 1 + 1
[1,2] => [2]
=> []
=> ?
=> ? = 1 + 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,3] => [3]
=> []
=> ?
=> ? = 1 + 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 + 1
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 1 + 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 6 + 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? = 1 + 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 6 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? = 6 + 1
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Values
[1] => [1]
=> []
=> ?
=> ? = 1 + 1
[1,2] => [2]
=> []
=> ?
=> ? = 1 + 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,3] => [3]
=> []
=> ?
=> ? = 1 + 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 + 1
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 1 + 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 6 + 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? = 1 + 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 6 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? = 6 + 1
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 75%●distinct values known / distinct values provided: 8%
Values
[1] => [1]
=> []
=> ?
=> ? = 1 + 1
[1,2] => [2]
=> []
=> ?
=> ? = 1 + 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,3] => [3]
=> []
=> ?
=> ? = 1 + 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 1 + 1
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 + 1
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 1 + 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 1 + 1
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2 + 1
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 + 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 6 + 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? = 1 + 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 + 1
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 6 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 + 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? = 1 + 1
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? = 2 + 1
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? = 6 + 1
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St000264
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Values
[1] => ([],1)
=> [1] => ([],1)
=> ? = 1 + 3
[1,2] => ([],2)
=> [2] => ([],2)
=> ? = 1 + 3
[2,1] => ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> ? = 1 + 3
[1,2,3] => ([],3)
=> [3] => ([],3)
=> ? = 1 + 3
[1,3,2] => ([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[2,1,3] => ([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[2,3,1] => ([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,3,4] => ([],4)
=> [4] => ([],4)
=> ? = 1 + 3
[1,2,4,3] => ([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[1,3,2,4] => ([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,1,3,4] => ([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 6 + 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,3,4,5] => ([],5)
=> [5] => ([],5)
=> ? = 1 + 3
[1,2,3,5,4] => ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,2,4,3,5] => ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,2,4,5] => ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 6 + 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,3,4,5] => ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 6 + 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001570
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => ([],2)
=> ([],1)
=> ? = 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
[1,2,3] => ([],3)
=> ([],1)
=> ? = 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 6
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 6
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 6
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001629
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 8% ●values known / values provided: 55%●distinct values known / distinct values provided: 8%
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 8% ●values known / values provided: 55%●distinct values known / distinct values provided: 8%
Values
[1] => ([],1)
=> ([],1)
=> [1] => ? = 1
[1,2] => ([],2)
=> ([],1)
=> [1] => ? = 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,2,3] => ([],3)
=> ([],1)
=> [1] => ? = 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,2,3,4] => ([],4)
=> ([],1)
=> [1] => ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 6
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1] => ? = 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 6
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 6
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001603
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 55%●distinct values known / distinct values provided: 8%
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 55%●distinct values known / distinct values provided: 8%
Values
[1] => ([],1)
=> ([],1)
=> [1]
=> ? = 1 + 1
[1,2] => ([],2)
=> ([],1)
=> [1]
=> ? = 1 + 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,2,3] => ([],3)
=> ([],1)
=> [1]
=> ? = 1 + 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1]
=> ? = 1 + 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 6 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1]
=> ? = 1 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 6 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 6 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph.
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