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Your data matches 254 different statistics following compositions of up to 3 maps.
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Matching statistic: St000768
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,1] => 0 = 1 - 1
[2] => 0 = 1 - 1
[1,1,1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[3] => 0 = 1 - 1
[1,1,1,1] => 0 = 1 - 1
[1,1,2] => 0 = 1 - 1
[1,2,1] => 1 = 2 - 1
[1,3] => 0 = 1 - 1
[2,1,1] => 0 = 1 - 1
[2,2] => 0 = 1 - 1
[3,1] => 0 = 1 - 1
[4] => 0 = 1 - 1
[1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => 0 = 1 - 1
[1,1,2,1] => 1 = 2 - 1
[1,1,3] => 0 = 1 - 1
[1,2,1,1] => 1 = 2 - 1
[1,2,2] => 0 = 1 - 1
[1,3,1] => 1 = 2 - 1
[1,4] => 0 = 1 - 1
[2,1,1,1] => 0 = 1 - 1
[2,1,2] => 0 = 1 - 1
[2,2,1] => 0 = 1 - 1
[2,3] => 0 = 1 - 1
[3,1,1] => 0 = 1 - 1
[3,2] => 0 = 1 - 1
[4,1] => 0 = 1 - 1
[5] => 0 = 1 - 1
Description
The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St000036
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 1
[1,1] => [1,0,1,0]
=> [2,1] => [2,1] => 1
[2] => [1,1,0,0]
=> [1,2] => [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => [2,1,3] => 1
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1,2,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,1,4,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,1,5,3,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,4,1,5] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,5,3] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,1,2,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,2,4,5] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,1,5,2,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,4,2,5] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1,2,3,5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,4,2,3,5] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001257
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001493
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001941
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001941: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001941: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1
Description
The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.3]) with parameters given by the identity and the permutation.
Also the number of paths in the Bruhat order from the identity to the permutation that are increasing with respect to a given reflection ordering as defined in Björner and Brenti [1, Section 5.2].
Matching statistic: St000034
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000664
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2] => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3] => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[1,1,2,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0 = 1 - 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,2,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1 = 2 - 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0 = 1 - 1
[2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0 = 1 - 1
[3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0 = 1 - 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0 = 1 - 1
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000966
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
Number of peaks minus the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,1] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
The following 244 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001411The number of patterns 321 or 3412 in a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000408The number of occurrences of the pattern 4231 in a permutation. St000842The breadth of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000068The number of minimal elements in a poset. St000699The toughness times the least common multiple of 1,. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001768The number of reduced words of a signed permutation. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000264The girth of a graph, which is not a tree. St001722The number of minimal chains with small intervals between a binary word and the top element. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000284The Plancherel distribution on integer partitions. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000694The number of affine bounded permutations that project to a given permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000729The minimal arc length of a set partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers 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. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000039The number of crossings of a permutation. St000058The order of a permutation. St000084The number of subtrees. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000478Another weight of a partition according to Alladi. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000872The number of very big descents of a permutation. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000219The number of occurrences of the pattern 231 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001645The pebbling number of a connected graph. St001933The largest multiplicity of a part in an integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.
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