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Matching statistic: St001663
St001663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[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] => 1
[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] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[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 number of occurrences of the Hertzsprung pattern 132 in a permutation.
A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.
Matching statistic: St001198
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 + 2
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,1} + 2
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,1} + 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,5,4,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,1,3,2,5] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,2,5,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,4,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,4,6,5,1] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,6,5,1,4] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,4,3,6,5,1] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 + 2
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,1} + 2
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,1} + 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1} + 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[2,5,4,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,1,3,2,5] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,2,5,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,4,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,4,6,5,1] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,3,6,5,1,4] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,4,3,6,5,1] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 2
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 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: St000205
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000621
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,5,3,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,5,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,5,2,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,5,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,5,4,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,5,3,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,2,5,3,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,3,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[5,2,1,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[5,3,1,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5,3,6,4] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,4,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,5,4,6] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000938
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [3,2]
=> [2]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,2,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,5,4,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,3,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,5,3,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,2,1,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,1,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,5,3,6,4] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,5,4,6] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
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 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [3,2]
=> [2]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,2,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,5,4,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,3,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,5,3,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,2,1,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,1,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5,3,6,4] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,4,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,5,4,6] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
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 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [3,2]
=> [2]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,2,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,5,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,5,4,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,3,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,5,3,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,2,1,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,1,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5,3,6,4] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,4,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,5,4,6] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
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: St001175
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,1}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,1}
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> 0
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,5,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,2,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2,5] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,4,1,2,3] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,4,1,6,5] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,1,4,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,1,6,4] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,6,1,4,5] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,6,1,5,4] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,3,1,5,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,3,1,6,5] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,5,1,3,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,5,1,6,3] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,6,1,3,5] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,6,1,5,3] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,3,1,4,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,3,1,6,4] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,4,1,3,6] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,4,1,6,3] => [2,6,5,1,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
The following 154 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000379The number of Hamiltonian cycles in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001498The normalised height of a Nakayama algebra with magnitude 1. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000929The constant term of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000941The number of characters of the symmetric group whose value on the partition is even. St001651The Frankl number of a lattice. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000699The toughness times the least common multiple of 1,. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000781The number of proper colouring schemes of a Ferrers diagram. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St001820The size of the image of the pop stack sorting operator. St001330The hat guessing number of a graph. St000648The number of 2-excedences of a permutation. St001846The number of elements which do not have a complement in the lattice. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001720The minimal length of a chain of small intervals in a lattice. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001845The number of join irreducibles minus the rank of a lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset.
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