Your data matches 463 different statistics following compositions of up to 3 maps.
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St000331: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> 1 = 3 - 2
Description
The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≄ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 2 - 2
[1,0,1,0]
=> [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,3,1] => 2 = 4 - 2
[1,1,1,0,0,0]
=> [3,2,1] => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3 = 5 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3 = 5 - 2
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 5
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000969
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 2
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
Description
We 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}$. Then we calculate the global dimension of that CNakayama algebra.
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 2
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
Description
The number of elements generated by the Dyck path as a map in the full transformation monoid. We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,4,5,1,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2,1,5,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,2,5,4] => 3 = 4 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,3,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,4,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,4,2,1] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 4 = 5 - 1
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,3,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,4,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,4,2,1] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 4 = 5 - 1
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
Description
The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,1,4,2,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,1,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,3,4,1,2] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,1,2,5,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,1,4,2,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,1,6,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,1,4,2,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,3,1,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,3,1,5,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,6,1,2,3,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,6,1,2,3,4] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,4,5,1,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,3,1,4,2,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,3,1,2,6,4] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,6,1,2,4,3] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,3,1,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,2,1,5,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,2,4,1,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,2,6,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,2,4,1,5,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,1,6,5,2,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [6,1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [5,3,6,1,2,4] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,4,1,6,2,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,3,4,1,5,2] => 2 = 3 - 1
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
The following 453 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000021The number of descents of a permutation. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000203The number of external nodes of a binary tree. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000010The length of the partition. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St000299The number of nonisomorphic vertex-induced subtrees. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000507The number of ascents of a standard tableau. St000619The number of cyclic descents of a permutation. St000670The reversal length of a permutation. St000912The number of maximal antichains in a poset. St001135The projective dimension of the first simple module in the Nakayama algebra 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. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001530The depth of a Dyck path. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000157The number of descents of a standard tableau. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001176The size of a partition minus its first part. St001388The number of non-attacking neighbors of a permutation. St001427The number of descents of a signed permutation. St001812The biclique partition number of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000925The number of topologically connected components of a set partition. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000809The reduced reflection length of the permutation. St001668The number of points of the poset minus the width of the poset. St000291The number of descents of a binary word. St001432The order dimension of the partition. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000336The leg major index of a standard tableau. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000741The Colin de VerdiĆØre graph invariant. St001889The size of the connectivity set of a signed permutation. St001864The number of excedances of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001863The number of weak excedances of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000628The balance of a binary word. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001330The hat guessing number of a graph. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000259The diameter of a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001782The order of rowmotion on the set of order ideals of a poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St000080The rank of the poset. St000260The radius of a connected graph. St001596The number of two-by-two squares inside a skew partition. St000451The length of the longest pattern of the form k 1 2. St000381The largest part of an integer composition. St000643The size of the largest orbit of antichains under Panyushev complementation. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001497The position of the largest weak excedence of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000624The normalized sum of the minimal distances to a greater element. St000455The second largest eigenvalue of a graph if it is integral. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001960The number of descents of a permutation minus one if its first entry is not one. St000808The number of up steps of the associated bargraph. St000956The maximal displacement of a permutation. St000039The number of crossings of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001390The number of bumps occurring when Schensted-inserting the letter 1 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. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001948The number of augmented double ascents of a permutation. St000120The number of left tunnels of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001712The number of natural descents of a standard Young tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000907The number of maximal antichains of minimal length in a poset. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000493The los statistic of a set partition. St000497The lcb statistic of a set partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000264The girth of a graph, which is not a tree. St000891The number of distinct diagonal sums of a permutation matrix. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000237The number of small exceedances. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001115The number of even descents of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001488The number of corners of a skew partition. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001435The number of missing boxes in the first row. St000031The number of cycles in the cycle decomposition of a permutation. St000919The number of maximal left branches of a binary tree. St000366The number of double descents of a permutation. St000665The number of rafts of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001083The number of boxed occurrences of 132 in a permutation. St001394The genus of a permutation. St000098The chromatic number of a graph. St000444The length of the maximal rise of a Dyck path. St000982The length of the longest constant subword. St000141The maximum drop size of a permutation. St000389The number of runs of ones of odd length in a binary word. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000441The number of successions of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001060The distinguishing index of a graph. St001581The achromatic number of a graph. St000306The bounce count of a Dyck path. St001096The size of the overlap set of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001090The number of pop-stack-sorts needed to sort a permutation. St000359The number of occurrences of the pattern 23-1. St000534The number of 2-rises of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001875The number of simple modules with projective dimension at most 1. St000013The height of a Dyck path. St000071The number of maximal chains in a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000201The number of leaf nodes in a binary tree. St001052The length of the exterior of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001893The flag descent of a signed permutation. St000356The number of occurrences of the pattern 13-2. St000647The number of big descents of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000527The width of the poset. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. 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 $eA$. 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 $eA$. 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 $eA$. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001153The number of blocks with even minimum in a set partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001867The number of alignments of type EN of a signed permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000308The height of the tree associated to a permutation. St000502The number of successions of a set partitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001737The number of descents of type 2 in a permutation. St000023The number of inner peaks of a permutation. St000174The flush statistic of a semistandard tableau. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000663The number of right floats of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St000015The number of peaks of a Dyck path. St000172The Grundy number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000746The number of pairs with odd minimum in a perfect matching. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000904The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St000942The number of critical left to right maxima of the parking functions. St000983The length of the longest alternating subword. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama 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: 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. St001261The Castelnuovo-Mumford regularity of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001806The upper middle entry of a permutation. St001902The number of potential covers of a poset. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000209Maximum difference of elements in cycles. St000238The number of indices that are not small weak excedances. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000272The treewidth of a graph. St000307The number of rowmotion orbits of a poset. St000314The number of left-to-right-maxima of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000387The matching number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000536The pathwidth 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. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000831The number of indices that are either descents or recoils. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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 $eA$. St001199The dominant 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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. St001512The minimum rank of a graph. St001524The degree of symmetry of a binary word. St001589The nesting number of a perfect matching. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001716The 1-improper chromatic number of a graph. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001807The lower middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001866The nesting alignments of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001884The number of borders of a binary word. St001928The number of non-overlapping descents in a permutation. St000004The major index of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000072The number of circled entries. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. 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. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000640The rank of the largest boolean interval in a poset. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000732The number of double deficiencies of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000794The mak of a permutation. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000961The shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001726The number of visible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001871The number of triconnected components of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001399The distinguishing number of a poset. St001644The dimension of a graph. St000677The standardized bi-alternating inversion number of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001638The book thickness of a graph.