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Your data matches 133 different statistics following compositions of up to 3 maps.
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Matching statistic: St001515
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
St001515: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[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]
=> 2
[1,1,1,0,0,0]
=> 3
[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]
=> 2
[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]
=> 3
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 4
[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]
=> 3
[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]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 4
[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]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 3
[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]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 4
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000672
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 3 = 4 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000325
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 4
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 4
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 3 = 4 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000007
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,5,4,3,2] => 4
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000015
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [[1]]
=> [1,0]
=> 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [[1,0],[0,1]]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [[0,1],[1,0]]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
The number of peaks of a Dyck path.
Matching statistic: St000213
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => [1] => 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [3,2,1] => 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => 2
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => 3
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [3,2,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => 3
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => 3
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,3,4] => 3
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [4,3,2,1] => 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => 3
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,3,4] => 3
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [2,3,4,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,3,5,4,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [2,3,4,1,5] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [2,3,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,4,3,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,3,5,4,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [2,3,4,1,5] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [2,3,1,4,5] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,5,4,3,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [2,3,1,4,5] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [3,2,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [3,2,4,1,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [3,2,5,4,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [3,2,4,1,5] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => [3,2,1,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,4,3,5,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,3,5,4,1] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [2,3,4,1,5] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [2,3,1,4,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,5,4,3,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [2,3,1,4,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [2,1,3,4,5] => 4
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000393
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 11 => 2
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 2
[1,1,0,1,0,0]
=> [3] => 100 => 001 => 2
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 3
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1001 => 3
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 3
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00111 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01001 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 10001 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 10001 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 00001 => 4
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000553
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000553: 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
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
The following 123 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000619The number of cyclic descents of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001668The number of points of the poset minus the width of the poset. St001875The number of simple modules with projective dimension at most 1. 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. St001083The number of boxed occurrences of 132 in a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000862The number of parts of the shifted shape of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001626The number of maximal proper sublattices of a lattice. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001863The number of weak excedances of a signed permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001394The genus of a permutation. St001488The number of corners of a skew partition. St000308The height of the tree associated to a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001645The pebbling number of a connected graph. St000035The number of left outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. 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 $eA$. St000356The number of occurrences of the pattern 13-2. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St001470The cyclic holeyness of a permutation. St001566The length of the longest arithmetic progression in a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000488The number of cycles of a permutation of length at most 2. St000539The number of odd inversions of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001405The number of bonds in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001962The proper pathwidth of a graph. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000236The number of cyclical small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000291The number of descents of a binary word. St000353The number of inner valleys of a permutation. St000390The number of runs of ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000562The number of internal points of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000872The number of very big descents of a permutation. St001471The magnitude of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001806The upper middle entry of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000735The last entry on the main diagonal of a standard tableau. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St001638The book thickness of a graph. St001812The biclique partition number of a graph.
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