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Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St001170
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
St001170: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
Description
Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 0 => 1
[1,1,0,0]
=> [1,2] => 1 => 0
[1,0,1,0,1,0]
=> [3,2,1] => 00 => 2
[1,0,1,1,0,0]
=> [2,3,1] => 00 => 2
[1,1,0,0,1,0]
=> [3,1,2] => 00 => 2
[1,1,0,1,0,0]
=> [2,1,3] => 01 => 2
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 001 => 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 001 => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 001 => 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 011 => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001958
(load all 88 compositions to match this statistic)
(load all 88 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [3,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 3 = 2 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,4,1,2,3] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,1,2,5,3] => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,5,2,3] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,1,5,4,2] => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,3,1,5,2] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 1 = 0 + 1
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St000019
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [4,3,1,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000209
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000225
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [4,3,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [4,3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [4,2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [4,3,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [4,1,1,1,1]
=> 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000653
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,1,3,2] => [4,2,3,1] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => [4,2,3,1] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,1,2] => [4,3,2,1] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4,2,3,1] => [4,3,2,1] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => [4,3,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,4,2,1] => [4,3,2,1] => 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 3
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000956
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,4,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,4,2,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,4,2,1] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001207
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [4,3,1,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001227
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001332The number of steps on the non-negative side of the walk associated with the permutation. St000060The greater neighbor of the maximum. St000501The size of the first part in the decomposition of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000844The size of the largest block in the direct sum decomposition of a permutation. St001267The length of the Lyndon factorization of the binary word. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001437The flex of a binary word. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000259The diameter of a connected graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St000849The number of 1/3-balanced pairs in a poset. St000670The reversal length of a permutation. St000356The number of occurrences of the pattern 13-2. St000703The number of deficiencies of a permutation. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000632The jump number of the poset. St000374The number of exclusive right-to-left minima of a permutation. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000358The number of occurrences of the pattern 31-2. St000624The normalized sum of the minimal distances to a greater element. St000663The number of right floats of a permutation. St000800The number of occurrences of the vincular pattern |231 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000242The number of indices that are not cyclical small weak excedances. St000354The number of recoils of a permutation. 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. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001489The maximum of the number of descents and the number of inverse descents. St001760The number of prefix or suffix reversals needed to sort a permutation. St001822The number of alignments of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001330The hat guessing number of a graph. St001623The number of doubly irreducible elements of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000307The number of rowmotion orbits of a poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001846The number of elements which do not have a complement in the lattice. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000617The number of global maxima of a Dyck path. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001820The size of the image of the pop stack sorting operator. St001618The cardinality of the Frattini sublattice of a lattice. St001626The number of maximal proper sublattices of a lattice. St001645The pebbling number of a connected graph.
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