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Your data matches 288 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 111 compositions to match this statistic)
(load all 111 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 2
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: St000542
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St001390
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 2
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
Matching statistic: St000864
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000864: 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] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 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] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 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] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St000011
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000066
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ 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]
=> [[1,0],[0,1]]
=> [[1,0],[0,1]]
=> 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [[0,1],[1,0]]
=> 2
[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
[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
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 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]]
=> 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]]
=> 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
[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
[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
[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
[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
[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]]
=> 2
[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]]
=> 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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
[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]]
=> 2
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St000068
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ 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]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
Description
The number of minimal elements in a poset.
Matching statistic: St000069
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ 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]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
Description
The number of maximal elements of a poset.
Matching statistic: St000314
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000314: 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]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000991
(load all 94 compositions to match this statistic)
(load all 94 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000991: 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]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,1,3] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,4,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,3,1,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,5,2,4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,5,1,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,4,1,5,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,5,3,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,5,2,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [4,2,5,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,2,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,1,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,1,3,5,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
The following 278 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St000535The rank-width of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000153The number of adjacent cycles of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000258The burning number of a graph. St000273The domination number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000397The Strahler number of a rooted tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000701The protection number of a binary tree. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000843The decomposition number of a perfect matching. St000916The packing number of a graph. St000918The 2-limited packing number of a graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001829The common independence number of a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000160The multiplicity of the smallest part of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000352The Elizalde-Pak rank of a permutation. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000546The number of global descents of a permutation. 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 $c_{n-1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001484The number of singletons of an integer partition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001826The maximal number of leaves on a vertex of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000906The length of the shortest maximal chain in a poset. St000990The first ascent of a permutation. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000552The number of cut vertices of a graph. St001691The number of kings in a graph. St000897The number of different multiplicities of parts of an integer partition. St000475The number of parts equal to 1 in a partition. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000374The number of exclusive right-to-left minima of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000392The length of the longest run of ones in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001728The number of invisible descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001060The distinguishing index of a graph. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000092The number of outer peaks of a permutation. St000456The monochromatic index of a connected graph. St000700The protection number of an ordered tree. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001948The number of augmented double ascents of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000781The number of proper colouring schemes of a Ferrers diagram. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001982The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000764The number of strong records in an integer composition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. 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 $eA$. St001870The number of positive entries followed by a negative entry in a signed permutation. St001975The corank of the alternating sign matrix. St001820The size of the image of the pop stack sorting operator. St001330The hat guessing number of a graph. St000891The number of distinct diagonal sums of a permutation matrix. St000640The rank of the largest boolean interval in a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001621The number of atoms of a lattice. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000873The aix statistic of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral.
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