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Your data matches 460 different statistics following compositions of up to 3 maps.
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Matching statistic: St001330
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001486
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2] => 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6] => 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6] => 2
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00272: Binary words —Gray next⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00272: Binary words —Gray next⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => 11 => 01 => 1 = 2 - 1
[1,1,0,0]
=> [2] => 10 => 11 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 101 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 010 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 001 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => 100 => 110 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => 100 => 110 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1100 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1100 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1100 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1100 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1100 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 01000 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00100 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 00010 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 00001 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 11000 => 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 110000 => 010000 => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => 101000 => 001000 => 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => 100100 => 000100 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => 100010 => 000010 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [6] => 100000 => 110000 => 1 = 2 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000651
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,3,4,5,6] => 1 = 2 - 1
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000100
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
Description
The number of linear extensions of a poset.
Matching statistic: St000661
(load all 60 compositions to match this statistic)
(load all 60 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St000931
(load all 60 compositions to match this statistic)
(load all 60 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001141
(load all 305 compositions to match this statistic)
(load all 305 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of occurrences of hills of size 3 in a Dyck path.
A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St000402
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 1 = 2 - 1
Description
Half the size of the symmetry class of a permutation.
The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]).
This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.
Matching statistic: St000529
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000529: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000529: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0 => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 01 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 00 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 11 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0000 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1111 => 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00000 => 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 00000 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 00000 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 1 = 2 - 1
Description
The number of permutations whose descent word is the given binary word.
This is the sizes of the preimages of the map [[Mp00109]].
The following 450 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000983The length of the longest alternating subword. St001128The exponens consonantiae of a partition. St001313The number of Dyck paths above the lattice path given by a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. 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. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001175The size of a partition minus the hook length of the base cell. St000629The defect of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000627The exponent of a binary word. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001586The number of odd parts smaller than the largest even part in an integer partition. St000657The smallest part of an integer composition. St000806The semiperimeter of the associated bargraph. St001432The order dimension of the partition. St001732The number of peaks visible from the left. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000369The dinv deficit of a Dyck path. St000655The length of the minimal rise of a Dyck path. St000478Another weight of a partition according to Alladi. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000618The number of self-evacuating tableaux of given shape. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. 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. St000877The depth of the binary word interpreted as a path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000667The greatest common divisor of the parts of 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 $eA$. 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 $eA$. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. 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. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000993The multiplicity of the largest part of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000296The length of the symmetric border of a binary word. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000439The position of the first down step of a Dyck path. St000306The bounce count of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St001371The length of the longest Yamanouchi prefix of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000145The Dyson rank of a partition. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000264The girth of a graph, which is not a tree. St000675The number of centered multitunnels of a Dyck path. St001910The height of the middle non-run of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000258The burning number of a graph. St000469The distinguishing number of a graph. St001580The acyclic chromatic number of a graph. St001746The coalition number of a graph. St001883The mutual visibility number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000778The metric dimension of a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001340The cardinality of a minimal non-edge isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001792The arboricity of a graph. St001949The rigidity index of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St001962The proper pathwidth of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000379The number of Hamiltonian cycles in a graph. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000026The position of the first return of a Dyck path. St000456The monochromatic index of a connected graph. St000485The length of the longest cycle of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St000354The number of recoils of a permutation. St000376The bounce deficit of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000699The toughness times the least common multiple of 1,. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001281The normalized isoperimetric number of a graph. St000886The number of permutations with the same antidiagonal sums. St000353The number of inner valleys of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. 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. St000836The number of descents of distance 2 of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000053The number of valleys of the Dyck path. St000260The radius of a connected graph. 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. St001118The acyclic chromatic index of a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000219The number of occurrences of the pattern 231 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000455The second largest eigenvalue of a graph if it is integral. St000961The shifted major index of a permutation. St001545The second Elser number of a connected graph. St001060The distinguishing index of a graph. St001481The minimal height of a peak of a Dyck path. St000477The weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001130The number of two successive successions in a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000517The Kreweras number of an integer partition. St000913The number of ways to refine the partition into singletons. 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,. St000999Number of indecomposable projective module with injective 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. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001159Number of simple modules with dominant dimension equal to the global dimension 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 $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding 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. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001082The number of boxed occurrences of 123 in a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000640The rank of the largest boolean interval in a poset. St000236The number of cyclical small weak excedances. St000171The degree of the graph. St000271The chromatic index of a graph. St000822The Hadwiger number of the graph. St001112The 3-weak dynamic number of a graph. St001642The Prague dimension of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001731The factorization defect of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000741The Colin de Verdière graph invariant. St001220The width of a permutation. St000216The absolute length of a permutation. St000837The number of ascents of distance 2 of a permutation. St001388The number of non-attacking neighbors of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000045The number of linear extensions of a binary tree. St000464The Schultz index of a connected graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000015The number of peaks of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001530The depth of a Dyck path. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000117The number of centered tunnels of a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000635The number of strictly order preserving maps of a poset into itself. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St000461The rix statistic of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000259The diameter of a connected graph. St000873The aix statistic of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001777The number of weak descents in an integer composition. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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 $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition.
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