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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000500
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000500: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 4
[2,1] => 0
[1,2,3] => 9
[1,3,2] => 4
[2,1,3] => 0
[2,3,1] => 4
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 16
[1,2,4,3] => 10
[1,3,2,4] => 6
[1,3,4,2] => 10
[1,4,2,3] => 6
[1,4,3,2] => 6
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 6
[2,3,4,1] => 10
[2,4,1,3] => 4
[2,4,3,1] => 6
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 2
[3,2,4,1] => 0
[3,4,1,2] => 4
[3,4,2,1] => 6
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 2
[4,2,3,1] => 0
[4,3,1,2] => 2
[4,3,2,1] => 0
[1,2,3,4,5] => 25
[1,2,3,5,4] => 18
[1,2,4,3,5] => 14
[1,2,4,5,3] => 18
[1,2,5,3,4] => 14
[1,2,5,4,3] => 13
[1,3,2,4,5] => 8
[1,3,2,5,4] => 7
[1,3,4,2,5] => 14
[1,3,4,5,2] => 18
[1,3,5,2,4] => 11
[1,3,5,4,2] => 13
[1,4,2,3,5] => 8
[1,4,2,5,3] => 7
[1,4,3,2,5] => 9
[1,4,3,5,2] => 7
[1,4,5,2,3] => 11
Description
Eigenvalues of the random-to-random operator acting on the regular representation.
This statistic is defined for a permutation $w$ as:
$$
\left[\binom{\ell(w) + 1}{2} + \operatorname{diag}\left(Q(w)\right)\right]
-
\left[\binom{\ell(u) + 1}{2} + \operatorname{diag}\left(Q(u)\right)\right]
$$
where:
* $u$ is the longest suffix of $w$ (viewed as a word) whose first ascent is even;
* $\ell(w)$ is the size of the permutation $w$ (equivalently, the length of the word $w$);
* $Q(w), Q(u)$ denote the recording tableaux of $w, u$ under the RSK correspondence;
* $\operatorname{diag}(\lambda)$ denotes the ''diagonal index'' (or ''content'') of an integer partition $\lambda$;
* and $\operatorname{diag}(T)$ of a tableau $T$ denotes the diagonal index of the partition given by the shape of $T$.
The regular representation of the symmetric group of degree n has dimension n!, so any linear operator acting on this vector space has n! eigenvalues (counting multiplicities). Hence, the eigenvalues of the random-to-random operator can be indexed by permutations; and the values of this statistic give all the eigenvalues of the operator (Theorem 12 of [1]).
Matching statistic: St000508
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000508: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> ? = 1
[1,2] => [[1,2]]
=> 4
[2,1] => [[1],[2]]
=> 0
[1,2,3] => [[1,2,3]]
=> 9
[1,3,2] => [[1,2],[3]]
=> 4
[2,1,3] => [[1,3],[2]]
=> 0
[2,3,1] => [[1,2],[3]]
=> 4
[3,1,2] => [[1,3],[2]]
=> 0
[3,2,1] => [[1],[2],[3]]
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> 16
[1,2,4,3] => [[1,2,3],[4]]
=> 10
[1,3,2,4] => [[1,2,4],[3]]
=> 6
[1,3,4,2] => [[1,2,3],[4]]
=> 10
[1,4,2,3] => [[1,2,4],[3]]
=> 6
[1,4,3,2] => [[1,2],[3],[4]]
=> 6
[2,1,3,4] => [[1,3,4],[2]]
=> 0
[2,1,4,3] => [[1,3],[2,4]]
=> 0
[2,3,1,4] => [[1,2,4],[3]]
=> 6
[2,3,4,1] => [[1,2,3],[4]]
=> 10
[2,4,1,3] => [[1,2],[3,4]]
=> 4
[2,4,3,1] => [[1,2],[3],[4]]
=> 6
[3,1,2,4] => [[1,3,4],[2]]
=> 0
[3,1,4,2] => [[1,3],[2,4]]
=> 0
[3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> 0
[3,4,1,2] => [[1,2],[3,4]]
=> 4
[3,4,2,1] => [[1,2],[3],[4]]
=> 6
[4,1,2,3] => [[1,3,4],[2]]
=> 0
[4,1,3,2] => [[1,3],[2],[4]]
=> 0
[4,2,1,3] => [[1,4],[2],[3]]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> 0
[4,3,1,2] => [[1,4],[2],[3]]
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 25
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 18
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 14
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 18
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> 14
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 13
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 8
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 7
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> 14
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 18
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> 11
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 13
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> 8
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> 7
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 9
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 7
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 11
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 13
Description
Eigenvalues of the random-to-random operator acting on a simple module.
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module [1].
Matching statistic: St001604
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Values
[1] => [] => []
=> ?
=> ? = 1
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,4}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,4}
[1,2,3] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,3,4,5] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[4,3,2,1,5] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,2,5,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,5,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,5,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[5,4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2,6] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,6,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,6,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,6,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,6,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,5,4,3,6] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,4,6,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,6,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,6,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,4,3,6] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,4,6,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,6,4,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,1,3,6] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,1,6,3] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,3,1,6] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,3,6,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,6,1,3] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,6,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,6,1,4,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,6,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,6,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,6,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,6,5,1,4,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,6,5,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,6,5,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,5,4,2,6] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,1,5,4,6,2] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,1,5,6,4,2] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,1,6,5,4,2] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,2,1,5,4,6] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,1,5,6,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,1,6,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,1,4,6] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,1,6,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,4,1,6] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,4,6,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,6,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,6,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,2,6,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,6,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,6,5,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,4,2,6] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,4,6,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,6,4,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,2,1,4,6] => [3,5,2,1,4] => [2,2,1]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000936
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,4}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,4}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,1,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,1,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,6,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,6,4,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,2,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,6,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,4,6,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,6,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,6,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,6,4,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,1,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,3,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,1,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,5,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,1,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,5,3,1,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,3,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,4}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,4}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,1,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,1,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,6,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,6,4,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,2,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,6,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,4,6,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,6,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,6,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,6,4,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,1,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,3,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,1,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,5,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,1,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,5,3,1,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,3,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000940
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,4}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,4}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,1,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,1,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,6,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,6,4,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,2,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,6,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,4,6,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,6,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,6,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,6,4,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,1,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,3,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,1,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,5,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,1,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,5,3,1,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,3,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St001124
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 29%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,4}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,4}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,1,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,1,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,2,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,6,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,6,4,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,4,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,3,4,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,2,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,6,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,4,6,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,3,6,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,4,6,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,5,6,4,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,1,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,3,5,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,1,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,3,5,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,4,5,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,1,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,6,5,3,1,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,6,5,3,4,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001570
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,4}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,4}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,2,4,3] => [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] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(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: St001645
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,4}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,4}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,9}
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,9}
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,9}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,9}
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,6,6,6,6,6,6,10,10,10,16}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,1,2,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,2,1,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,3,6,4,2] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,4,1,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,4,3,5,6,1] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,1,6,4,3] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,3,6,4,1] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,1,4,6,2,5] => [1,3,4,6,5,2] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[3,1,6,2,4,5] => [1,3,6,5,4,2] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,2,4,6,1,5] => [1,3,4,6,5,2] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[3,2,6,1,4,5] => [1,3,6,5,4,2] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,1,2,5,6,3] => [1,4,5,6,3,2] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,1,2,6,3,5] => [1,4,6,5,3,2] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,1,5,6,2,3] => [1,4,6,3,5,2] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,1,6,3,2,5] => [1,4,3,6,5,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,2,1,5,6,3] => [1,4,5,6,3,2] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,2,1,6,3,5] => [1,4,6,5,3,2] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,2,5,6,1,3] => [1,4,6,3,5,2] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,2,6,3,1,5] => [1,4,3,6,5,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,1,2,3,6,4] => [1,5,6,4,3,2] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The pebbling number of a connected graph.
Matching statistic: St000264
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,4}
[2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,4}
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,4,4,9}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,4,4,9}
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,4,4,9}
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,4,4,9}
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,4,4,9}
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,4,4,9}
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,1,4] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,2,4] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,1,2] => [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,1,3] => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,2,3,1] => [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,1,2] => [4,3,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,4,4,6,6,6,6,6,6,10,10,10,16}
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,13,14,14,14,14,18,18,18,18,25}
[2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
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