Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001381
(load all 14 compositions to match this statistic)
Mapping
St001381: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 0
[1,2,3] => 5
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 14
[1,2,4,3] => 0
[1,3,2,4] => 2
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 4
[2,1,4,3] => 0
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 42
[1,2,3,5,4] => 0
[1,2,4,3,5] => 5
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 9
[1,3,2,5,4] => 0
[1,3,4,2,5] => 5
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 4
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The fertility of a permutation. This is the size of the preimage of a permutation $\pi$ under the stack-sorting map $s$ defined by Julian West. For a partial permutation $\pi = \pi_1 \cdots \pi_n$ define $s(\pi)$ recursively as follows: * If the $n\leq 1$ set $s(\pi) = \pi$. * Otherwise, let $m$ be the largest entry in $\pi$, write $\pi=LmR$ and define $s(\pi) = s(L)s(R)m$.
Matching statistic: St001199
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1 + 1
[1,2] => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2 + 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 5 + 1
[1,3,2] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 0 + 1
[2,1,3] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? = 1 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[3,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 14 + 1
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,4,2,1] => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[4,1,3,2] => [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4,3,1,2] => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 42 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,4,2,3] => [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 14 + 1
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 1
[2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 1
[3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,4,2,1,5] => [5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[4,1,3,2,5] => [5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[4,2,3,1,5] => [5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 132 + 1
[1,2,3,5,4,6] => [6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 1
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 1
[1,2,4,5,3,6] => [6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 1
[1,2,5,3,4,6] => [6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 1
[1,2,5,4,3,6] => [6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 34 + 1
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 1
[1,3,4,5,2,6] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 1
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 1
[1,3,5,4,2,6] => [6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 20 + 1
[1,4,2,5,3,6] => [6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,4,3,2,5,6] => [6,5,2,3,4,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,4,3,5,2,6] => [6,2,5,3,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 63%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => ? = 2
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => ? = 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 5
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 0
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 0
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 0
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 14
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 0
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 42
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[1,5,4,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 14
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,4,1,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 0
[2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 0
[2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,5,1,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[2,5,4,1,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9
[3,1,2,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[3,1,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 0
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 0
[3,1,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,2,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,2,1,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[3,2,4,1,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 0
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 0
[3,2,5,4,1] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[3,4,5,1,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,4,5,2,1] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,5,1,2,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,5,1,4,2] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,5,2,1,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,5,2,4,1] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001198
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 60%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1 + 2
[1,2] => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2 + 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 0 + 2
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 5 + 2
[1,3,2] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? = 1 + 2
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[3,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 0 + 2
[3,2,1] => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 14 + 2
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 4 + 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 0 + 2
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[3,4,2,1] => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[4,1,3,2] => [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,3,1,2] => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 42 + 2
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,5,3,2] => [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 0 + 2
[1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,2,3] => [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 14 + 2
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[3,4,2,1,5] => [5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[4,1,3,2,5] => [5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[4,2,3,1,5] => [5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 132 + 2
[1,2,3,5,4,6] => [6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 2
[1,2,4,5,3,6] => [6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,2,5,3,4,6] => [6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 2
[1,2,5,4,3,6] => [6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 34 + 2
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 2
[1,3,4,5,2,6] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 2
[1,3,5,4,2,6] => [6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 20 + 2
[1,4,2,5,3,6] => [6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,4,3,2,5,6] => [6,5,2,3,4,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,4,3,5,2,6] => [6,2,5,3,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 60%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1 + 2
[1,2] => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2 + 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 0 + 2
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 5 + 2
[1,3,2] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? = 1 + 2
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[3,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 0 + 2
[3,2,1] => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 14 + 2
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 4 + 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 0 + 2
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[3,4,2,1] => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[4,1,3,2] => [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,3,1,2] => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 42 + 2
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,5,3,2] => [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 0 + 2
[1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,2,3] => [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 14 + 2
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 0 + 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 9 + 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[3,4,2,1,5] => [5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 2
[4,1,3,2,5] => [5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[4,2,3,1,5] => [5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 132 + 2
[1,2,3,5,4,6] => [6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 2
[1,2,4,5,3,6] => [6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,2,5,3,4,6] => [6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 2
[1,2,5,4,3,6] => [6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 34 + 2
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 24 + 2
[1,3,4,5,2,6] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 14 + 2
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10 + 2
[1,3,5,4,2,6] => [6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 20 + 2
[1,4,2,5,3,6] => [6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,4,3,2,5,6] => [6,5,2,3,4,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,4,3,5,2,6] => [6,2,5,3,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000456
(load all 7 compositions to match this statistic)
Mapping
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 34%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 1 + 1
[1,2] => [1,2] => ([],2)
 => ([],2)
 => ? = 2 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => ([],1)
 => ? = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ([],3)
 => ? = 5 + 1
[1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? = 1 + 1
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? = 0 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],4)
 => ? = 14 + 1
[1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? = 4 + 1
[2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 0 + 1
[2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? = 2 + 1
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 0 + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0 + 1
[3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? = 0 + 1
[4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],5)
 => ? = 42 + 1
[1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 5 + 1
[1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[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)
 => ([],1)
 => ? = 0 + 1
[1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 9 + 1
[1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 5 + 1
[1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 4 + 1
[1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 0 + 1
[1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 0 + 1
[1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 0 + 1
[1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,5,3,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,5,3,4,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[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)
 => ([],1)
 => ? = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 14 + 1
[2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0 + 1
[2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 9 + 1
[2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 5 + 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,5,1,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0 + 1
[2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 4 + 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0 + 1
[2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,4,5,1,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,5,1,4,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,3,1,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,5,3,4,1] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,1,2,5,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,4,5,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,5,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[3,2,1,5,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,4,2,5,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,5,2,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,5,2,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,3,5,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,2,3,5,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,5,1,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,5,2,1,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,5,2,3,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,1,3,4,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[5,2,3,1,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,4,1,3,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000936
Mapping
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: 24%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [2]
 => []
 => ?
 => ? = 2
[2,1] => [1,1]
 => [1]
 => []
 => ? = 0
[1,2,3] => [3]
 => []
 => ?
 => ? = 5
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 0
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 1
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 0
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 0
[3,2,1] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,3,4] => [4]
 => []
 => ?
 => ? = 14
[1,2,4,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,3,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,3,4] => [3,1]
 => [1]
 => []
 => ? = 4
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? = 2
[2,3,4,1] => [3,1]
 => [1]
 => []
 => ? = 0
[2,4,1,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,4,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,2,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,4,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,1,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[4,1,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5] => [5]
 => []
 => ?
 => ? = 42
[1,2,3,5,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,4,3,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,2,4,5,3] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,3,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2,4,5] => [4,1]
 => [1]
 => []
 => ? = 9
[1,3,2,5,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,4,2,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,3,5,2,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,3,5] => [4,1]
 => [1]
 => []
 => ? = 4
[1,4,2,5,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,5,2,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[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
Mapping
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: 24%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [2]
 => []
 => ?
 => ? = 2
[2,1] => [1,1]
 => [1]
 => []
 => ? = 0
[1,2,3] => [3]
 => []
 => ?
 => ? = 5
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 0
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 1
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 0
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 0
[3,2,1] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,3,4] => [4]
 => []
 => ?
 => ? = 14
[1,2,4,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,3,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,3,4] => [3,1]
 => [1]
 => []
 => ? = 4
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? = 2
[2,3,4,1] => [3,1]
 => [1]
 => []
 => ? = 0
[2,4,1,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,4,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,2,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,4,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,1,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[4,1,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5] => [5]
 => []
 => ?
 => ? = 42
[1,2,3,5,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,4,3,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,2,4,5,3] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,3,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2,4,5] => [4,1]
 => [1]
 => []
 => ? = 9
[1,3,2,5,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,4,2,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,3,5,2,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,3,5] => [4,1]
 => [1]
 => []
 => ? = 4
[1,4,2,5,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,5,2,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[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
Mapping
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: 24%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [2]
 => []
 => ?
 => ? = 2
[2,1] => [1,1]
 => [1]
 => []
 => ? = 0
[1,2,3] => [3]
 => []
 => ?
 => ? = 5
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 0
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 1
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 0
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 0
[3,2,1] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,3,4] => [4]
 => []
 => ?
 => ? = 14
[1,2,4,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,3,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,3,4] => [3,1]
 => [1]
 => []
 => ? = 4
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? = 2
[2,3,4,1] => [3,1]
 => [1]
 => []
 => ? = 0
[2,4,1,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,4,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,2,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,4,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,1,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[4,1,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5] => [5]
 => []
 => ?
 => ? = 42
[1,2,3,5,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,4,3,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,2,4,5,3] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,3,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2,4,5] => [4,1]
 => [1]
 => []
 => ? = 9
[1,3,2,5,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,4,2,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,3,5,2,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,3,5] => [4,1]
 => [1]
 => []
 => ? = 4
[1,4,2,5,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,5,2,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[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
Mapping
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: 24%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [2]
 => []
 => ?
 => ? = 2
[2,1] => [1,1]
 => [1]
 => []
 => ? = 0
[1,2,3] => [3]
 => []
 => ?
 => ? = 5
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 0
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 1
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 0
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 0
[3,2,1] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,3,4] => [4]
 => []
 => ?
 => ? = 14
[1,2,4,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,3,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,3,4] => [3,1]
 => [1]
 => []
 => ? = 4
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? = 2
[2,3,4,1] => [3,1]
 => [1]
 => []
 => ? = 0
[2,4,1,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,4,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? = 2
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,2,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,4,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,1,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[4,1,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5] => [5]
 => []
 => ?
 => ? = 42
[1,2,3,5,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,4,3,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,2,4,5,3] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,3,4] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2,4,5] => [4,1]
 => [1]
 => []
 => ? = 9
[1,3,2,5,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,4,2,5] => [4,1]
 => [1]
 => []
 => ? = 5
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,3,5,2,4] => [3,2]
 => [2]
 => []
 => ? = 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,3,5] => [4,1]
 => [1]
 => []
 => ? = 4
[1,4,2,5,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,5,2,3] => [3,2]
 => [2]
 => []
 => ? = 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[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.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.