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Your data matches 148 different statistics following compositions of up to 3 maps.
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Matching statistic: St000645
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> 4
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000957
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000957: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 4
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000327
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 4
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St000030
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [3,4,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => [3,2,4,1] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => [4,5,3,1,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => [5,4,2,3,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => [4,5,1,2,3] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => [5,4,2,1,3] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => [5,3,4,2,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => [3,4,5,1,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => [5,3,2,4,1] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => [5,3,2,1,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => [3,5,1,2,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [5,2,3,4,1] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => [5,2,3,1,4] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,5,2,4,1] => [4,5,3,2,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,4,1,5,2] => [5,4,3,1,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [4,5,2,3,1] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,1,5,2,3] => [5,4,1,2,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [4,5,2,1,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,3,2,4,1] => [4,3,5,2,1] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,3,2] => [3,5,4,1,2] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,4,1] => [4,3,2,5,1] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [3,4,1,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,4,1] => [4,2,3,5,1] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [3,5,4,2,1] => 4
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000081
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
Description
The number of edges of a graph.
Matching statistic: St000800
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [4,5,2,3,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [3,4,1,5,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [3,4,5,2,1] => 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [5,1,2,3,6,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => [5,1,2,6,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [5,1,2,6,3,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [4,1,2,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => [5,1,6,3,4,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => [4,1,5,3,6,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => [5,1,6,2,4,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => [5,1,6,2,3,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [4,1,5,2,6,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [3,1,5,6,4,2] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [4,1,5,6,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [4,1,5,6,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [3,1,4,5,6,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [5,6,2,3,4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [4,5,2,3,6,1] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [3,5,2,6,4,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [4,5,2,6,3,1] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [3,4,2,5,6,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [5,6,1,3,4,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => [4,5,1,3,6,2] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => [5,6,1,2,4,3] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,5,1,2,6,3] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [3,5,1,6,4,2] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,5,1,6,3,2] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,5,1,6,2,3] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [3,4,1,5,6,2] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [2,5,6,3,4,1] => 4
Description
The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000171
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,4,6,6,6}
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000987
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,4,6,6,6}
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001725
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,4,6,6,6,6} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6,6} + 1
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6,6} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6,6} + 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {4,4,6,6,6,6} + 1
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,4,6,6,6,6} + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St001232
(load all 113 compositions to match this statistic)
(load all 113 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 71%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4} + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4} + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,3,4,4} + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4} + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,5,5,5,5,5,5,5,6,6,6,6,6} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 138 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000456The monochromatic index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001822The number of alignments of a signed permutation. St000264The girth of a graph, which is not a tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001555The order of a signed permutation. St000662The staircase size of the code of a permutation. St001645The pebbling number of a connected graph. St000356The number of occurrences of the pattern 13-2. St000665The number of rafts of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000742The number of big ascents of a permutation after prepending zero. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000676The number of odd rises of a Dyck path. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001330The hat guessing number of a graph. St001769The reflection length of a signed permutation. St001198The 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 $eA$. St001206The 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 $eA$. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001083The number of boxed occurrences of 132 in a permutation. St001176The size of a partition minus its first part. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001961The sum of the greatest common divisors of all pairs of parts. St000670The reversal length of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000451The length of the longest pattern of the form k 1 2. St000422The energy of a graph, if it is integral. St000624The normalized sum of the minimal distances to a greater element. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St000956The maximal displacement of a permutation. St001115The number of even descents of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001875The number of simple modules with projective dimension at most 1. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000730The maximal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001863The number of weak excedances of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001420Half the length of a longest factor which is its own reverse-complement 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. St001439The number of even weak deficiencies and of odd weak exceedences. St000735The last entry on the main diagonal of a standard tableau. St000632The jump number of the poset. St001638The book thickness of a graph. St000307The number of rowmotion orbits of a poset. St000741The Colin de Verdière graph invariant. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice.
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