Your data matches 237 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001392
(load all 4 compositions to match this statistic)
Mapping
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0 = 2 - 2
[2]
 => 1 = 3 - 2
[1,1]
 => 0 = 2 - 2
[3]
 => 2 = 4 - 2
[2,1]
 => 0 = 2 - 2
[1,1,1]
 => 0 = 2 - 2
[4]
 => 3 = 5 - 2
[3,1]
 => 2 = 4 - 2
[2,2]
 => 1 = 3 - 2
[2,1,1]
 => 0 = 2 - 2
[1,1,1,1]
 => 0 = 2 - 2
[5]
 => 4 = 6 - 2
[4,1]
 => 3 = 5 - 2
[3,2]
 => 1 = 3 - 2
[3,1,1]
 => 2 = 4 - 2
[2,2,1]
 => 0 = 2 - 2
[2,1,1,1]
 => 0 = 2 - 2
[1,1,1,1,1]
 => 0 = 2 - 2
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000462
(load all 8 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 0 = 2 - 2
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 3 - 2
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 0 = 2 - 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 4 - 2
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 0 = 2 - 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0 = 2 - 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 3 = 5 - 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 2 = 4 - 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 3 - 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0 = 2 - 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 0 = 2 - 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => 4 = 6 - 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 3 = 5 - 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1 = 3 - 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 4 - 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 0 = 2 - 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0 = 2 - 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0 = 2 - 2
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000673
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 2
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [1,2,3,4,6,5] => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => 4
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [1,2,3,4,5,7,6] => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,6,5,2,3,4] => 5
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => 4
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 6
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St000724
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000724: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 2
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 3
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => 4
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [3,1,2] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,2,1,3] => 4
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,3,1,2] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => 5
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,2,3] => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => 4
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2
Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].
Matching statistic: St001463
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001463: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 3
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 6
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
Description
The number of distinct columns in the nullspace of a graph. Let $A$ be the adjacency matrix of a graph on $n$ vertices, and $K$ a $n\times d$ matrix whose column vectors form a basis of the nullspace of $A$. Then any other matrix $K'$ whose column vectors also form a basis of the nullspace is related to $K$ by $K' = K T$ for some invertible $d\times d$ matrix $T$. Any two rows of $K$ are equal if and only if they are equal in $K'$. The nullspace of a graph is usually written as a $d\times n$ matrix, hence the name of this statistic.
Matching statistic: St000018
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 3 = 4 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 3 = 4 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 4 = 5 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1 = 2 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [1,3,4,2] => 2 = 3 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => 4 = 5 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 5 = 6 - 1
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 4 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 5 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 6 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000154
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1 = 2 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 3 = 4 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4 = 5 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 1 = 2 - 1
Description
The sum of the descent bottoms of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$ For the descent tops, see [[St000111]].
Matching statistic: St000472
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,3,1] => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 3 = 4 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 4 = 5 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [6,5,4,3,1,2] => 1 = 2 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => 3 = 4 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 4 = 5 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [5,6,4,3,2,1] => 5 = 6 - 1
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000543
(load all 3 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 0 => 1 => 1 = 2 - 1
[2]
 => 0 => 1 => 0 => 1 = 2 - 1
[1,1]
 => 11 => 00 => 01 => 2 = 3 - 1
[3]
 => 1 => 0 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 10 => 11 => 1 = 2 - 1
[1,1,1]
 => 111 => 000 => 001 => 3 = 4 - 1
[4]
 => 0 => 1 => 0 => 1 = 2 - 1
[3,1]
 => 11 => 00 => 01 => 2 = 3 - 1
[2,2]
 => 00 => 11 => 00 => 1 = 2 - 1
[2,1,1]
 => 011 => 100 => 011 => 3 = 4 - 1
[1,1,1,1]
 => 1111 => 0000 => 0001 => 4 = 5 - 1
[5]
 => 1 => 0 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 10 => 11 => 1 = 2 - 1
[3,2]
 => 10 => 01 => 10 => 2 = 3 - 1
[3,1,1]
 => 111 => 000 => 001 => 3 = 4 - 1
[2,2,1]
 => 001 => 110 => 111 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 1000 => 0011 => 4 = 5 - 1
[1,1,1,1,1]
 => 11111 => 00000 => 00001 => 5 = 6 - 1
Description
The size of the conjugacy class of a binary word. Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
The following 227 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000626The minimal period of a binary word. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000988The orbit size of a permutation under Foata's bijection. St001090The number of pop-stack-sorts needed to sort a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001313The number of Dyck paths above the lattice path given by a binary word. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. 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. St000008The major index of the composition. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000355The number of occurrences of the pattern 21-3. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in 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. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000682The Grundy value of Welter's game on a binary word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001377The major index minus the number of inversions of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001841The number of inversions of a set partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000877The depth of the binary word interpreted as a path. St000054The first entry of the permutation. St000060The greater neighbor of the maximum. St000133The "bounce" of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000833The comajor index of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000654The first descent of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000454The largest eigenvalue of a graph if it is integral. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001556The number of inversions of the third entry of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000383The last part of an integer composition. St000456The monochromatic index of a connected graph. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000422The energy of a graph, if it is integral. St000438The position of the last up step in a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000770The major index of an integer partition when read from bottom to top. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000933The number of multipartitions of sizes given by an integer partition. St000981The length of the longest zigzag subpath. St001074The number of inversions of the cyclic embedding of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. 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 $eA$. St000014The number of parking functions supported by a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000294The number of distinct factors of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001814The number of partitions interlacing the given partition. St001885The number of binary words with the same proper border set. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000259The diameter of a connected graph. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000352The Elizalde-Pak rank of a permutation. St001645The pebbling number of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001545The second Elser number of a connected graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. 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. St000090The variation of a composition. St000091The descent variation of a composition. St000230Sum of the minimal elements of the blocks of a set partition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St001151The number of blocks with odd minimum. St001375The pancake length of a permutation. St001487The number of inner corners of a skew partition. St001516The number of cyclic bonds of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000365The number of double ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St001435The number of missing boxes in the first row. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000735The last entry on the main diagonal of a standard tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000307The number of rowmotion orbits of a poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000477The weight of a partition according to Alladi. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001118The acyclic chromatic index of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001959The product of the heights of the peaks of a Dyck path. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000260The radius of a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation.