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Your data matches 306 different statistics following compositions of up to 3 maps.
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Matching statistic: St000409
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(load all 11 compositions to match this statistic)
St000409: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0 = 1 - 1
[.,[.,.]]
=> 1 = 2 - 1
[[.,.],.]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> 1 = 2 - 1
[.,[[.,.],.]]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> 0 = 1 - 1
[[.,[.,.]],.]
=> 1 = 2 - 1
[[[.,.],.],.]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> 1 = 2 - 1
Description
The number of pitchforks in a binary tree.
A pitchfork is a subtree of a complete binary tree with exactly three leaves, see Section 3.2 of [1].
Matching statistic: St000665
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1,2] => 1 = 2 - 1
[[.,.],.]
=> [1,2] => [1,2] => 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,3,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2,3] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2,3] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,3,2,4] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,3,4] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,3,4] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3,4] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,3,2,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,3,4] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,3,4] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,2,3,4] => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St001613
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0 = 1 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001621
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0 = 1 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St000630
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [2,1] => 1 => 1
[.,[.,.]]
=> [1,0,1,0]
=> [3,1,2] => 10 => 2
[[.,.],.]
=> [1,1,0,0]
=> [2,3,1] => 10 => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 100 => 2
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 110 => 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 100 => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 101 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 100 => 2
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1000 => 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1010 => 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1100 => 2
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1100 => 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1100 => 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1000 => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1010 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1010 => 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1000 => 2
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1001 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1110 => 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1001 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1010 => 2
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1000 => 2
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St001720
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000143
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,1]
=> 1 = 2 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,1]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,3,2] => [2,1]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,4,2,3] => [3,1]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,3,4,2] => [3,1]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000257
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,1]
=> 1 = 2 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,1]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,3,2] => [2,1]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,4,2,3] => [3,1]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,3,4,2] => [3,1]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 2 - 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000845
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The maximal number of elements covering an element of a poset.
The following 296 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001568The smallest positive integer that does not appear twice in the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000741The Colin de Verdière graph invariant. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. 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. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. 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 $eA$. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St000920The logarithmic height of a Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000929The constant term of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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 $eA$. St001545The second Elser number of a connected graph. St001118The acyclic chromatic index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000454The largest eigenvalue of a graph if it is integral. St000661The number of rises of length 3 of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001691The number of kings in a graph. St000441The number of successions of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000552The number of cut vertices of a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000464The Schultz index of a connected graph. St000937The number of positive values 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. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001717The largest size of an interval in a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000080The rank of the poset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001668The number of points of the poset minus the width of the poset. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001117The game chromatic index of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001649The length of a longest trail in a graph. St001964The interval resolution global dimension of a poset. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000021The number of descents of a permutation. St000117The number of centered tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000245The number of ascents of a permutation. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000535The rank-width of a graph. St000619The number of cyclic descents of a permutation. St000644The number of graphs with given frequency partition. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000739The first entry in the last row of a semistandard tableau. St000742The number of big ascents of a permutation after prepending zero. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000958The number of Bruhat factorizations of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001281The normalized isoperimetric number of a graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001316The domatic number of a graph. St001323The independence gap of a graph. St001339The irredundance number of a graph. St001350Half of the Albertson index of a graph. St001358The largest degree of a regular subgraph of a graph. St001401The number of distinct entries in a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001489The maximum of the number of descents and the number of inverse descents. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001517The length of a longest pair of twins in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001589The nesting number of a perfect matching. St001637The number of (upper) dissectors of a poset. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001684The reduced word complexity of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001712The number of natural descents of a standard Young tableau. St001735The number of permutations with the same set of runs. St001739The number of graphs with the same edge polytope as the given graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001962The proper pathwidth of a graph. St000002The number of occurrences of the pattern 123 in a permutation. St000015The number of peaks of a Dyck path. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000144The pyramid weight of the Dyck path. St000223The number of nestings in the permutation. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000310The minimal degree of a vertex of a graph. St000325The width of the tree associated to a permutation. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000443The number of long tunnels of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000553The number of blocks of a graph. St000636The hull number of a graph. St000646The number of big ascents of a permutation. St000670The reversal length of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000991The number of right-to-left minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001271The competition number of a graph. St001286The annihilation number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001310The number of induced diamond graphs in a graph. St001315The dissociation number of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. 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. St001470The cyclic holeyness of a permutation. St001488The number of corners of a skew partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001513The number of nested exceedences of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001703The villainy of a graph. St001734The lettericity of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001765The number of connected components of the friends and strangers graph. St001783The number of odd automorphisms of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000172The Grundy number of a graph. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000451The length of the longest pattern of the form k 1 2. St000519The largest length of a factor maximising the subword complexity. St000537The cutwidth of a graph. St000922The minimal number such that all substrings of this length are unique. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001345The Hamming dimension of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001655The general position number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000637The length of the longest cycle in a graph. St000735The last entry on the main diagonal of a standard tableau.
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