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Your data matches 533 different statistics following compositions of up to 3 maps.
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Matching statistic: St001526
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
St001526: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001566
(load all 262 compositions to match this statistic)
(load all 262 compositions to match this statistic)
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
St001566: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001566: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Matching statistic: St001405
(load all 81 compositions to match this statistic)
(load all 81 compositions to match this statistic)
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St001405: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001405: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 4 - 1
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St000308
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00067: Permutations āFoata bijectionā¶ Permutations
St000308: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00067: Permutations āFoata bijectionā¶ Permutations
St000308: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => 2
[1,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,1,4,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,3,2] => 2
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => 3
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 3
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,2,4] => 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,3,5,1,2] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => 2
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,5,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,3,5,4,2] => 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 4
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000808
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000808: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000808: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1] => 1
[1,0,1,0]
=> [3,1,2] => [1,2] => 2
[1,1,0,0]
=> [2,3,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,3] => 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,2,1] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,2] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,2] => 2
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,4] => 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,2,2] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,2,2] => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,3] => 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,2] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,2,2] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1] => 4
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St000956
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000956: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00201: Dyck paths āRingelā¶ Permutations
St000956: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 2
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
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: St001039
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St001039: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St001039: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001090
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St001090: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00201: Dyck paths āRingelā¶ Permutations
St001090: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 2
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St001315
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St001315: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St001315: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4
Description
The dissociation number of a graph.
Matching statistic: St000628
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00093: Dyck paths āto binary wordā¶ Binary words
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000628: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000628: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 11 => 0 = 1 - 1
[1,0,1,0]
=> 1010 => 1101 => 1 = 2 - 1
[1,1,0,0]
=> 1100 => 1101 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 110101 => 1 = 2 - 1
[1,0,1,1,0,0]
=> 101100 => 110101 => 1 = 2 - 1
[1,1,0,0,1,0]
=> 110010 => 110101 => 1 = 2 - 1
[1,1,0,1,0,0]
=> 110100 => 111001 => 2 = 3 - 1
[1,1,1,0,0,0]
=> 111000 => 111001 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => 3 = 4 - 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
The following 523 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001096The size of the overlap set of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001949The rigidity index of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000171The degree of the graph. St000172The Grundy number of a graph. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000245The number of ascents of a permutation. St000271The chromatic index of a graph. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000831The number of indices that are either descents or recoils. St000904The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001118The acyclic chromatic index of a graph. St001119The length of a shortest maximal path in a graph. 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. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001486The number of corners of the ribbon associated with an integer composition. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001806The upper middle entry of a permutation. St001962The proper pathwidth of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000050The depth or height of a binary tree. St000080The rank of the poset. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000624The normalized sum of the minimal distances to a greater element. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000863The length of the first row of the shifted shape of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000982The length of the longest constant subword. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001557The number of inversions of the second entry of a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000326The position of the first one in a binary word after appending a 1 at the end. St000538The number of even inversions of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. 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 $c_{n-1}$. St000239The number of small weak excedances. St000382The first part of an integer composition. St000485The length of the longest cycle of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001461The number of topologically connected components of the chord diagram of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St001807The lower middle entry of a permutation. St001884The number of borders of a binary word. St000216The absolute length of a permutation. St000290The major index of a binary word. St000295The length of the border of a binary word. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000539The number of odd inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000691The number of changes of a binary word. St000732The number of double deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 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. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001388The number of non-attacking neighbors of a permutation. St001485The modular major index of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001811The Castelnuovo-Mumford regularity of a permutation. St001638The book thickness of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000654The first descent of a permutation. St001948The number of augmented double ascents 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 $eA$. St000489The number of cycles of a permutation of length at most 3. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000439The position of the first down step of a Dyck path. St000495The number of inversions of distance at most 2 of a permutation. St000676The number of odd rises of a Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000741The Colin de VerdiĆØre graph invariant. 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 $eA$. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000893The number of distinct diagonal sums of an alternating sign matrix. 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. St001060The distinguishing index of a graph. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001330The hat guessing number of a graph. St001644The dimension of a graph. St000259The diameter of a connected graph. St000444The length of the maximal rise of a Dyck path. St001863The number of weak excedances of a signed permutation. St000035The number of left outer peaks of a permutation. St000253The crossing number of a set partition. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000834The number of right outer peaks of a permutation. St000455The second largest eigenvalue of a graph if it is integral. 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. 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. St000518The number of distinct subsequences in a binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. 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. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. 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. 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000254The nesting number of a set partition. St000352The Elizalde-Pak rank of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000647The number of big descents of a permutation. St000665The number of rafts of a permutation. St000871The number of very big ascents of a permutation. St001394The genus of a permutation. St000264The girth of a graph, which is not a tree. St000806The semiperimeter of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001623The number of doubly irreducible elements of a lattice. 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. St000237The number of small exceedances. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001875The number of simple modules with projective dimension at most 1. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000058The order of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000307The number of rowmotion orbits of a poset. St000729The minimal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000921The number of internal inversions of a binary word. St000942The number of critical left to right maxima of the parking functions. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001062The maximal size of a block of a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001517The length of a longest pair of twins in a permutation. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001667The maximal size of a pair of weak twins for a permutation. St001691The number of kings in a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001937The size of the center of a parking function. St000023The number of inner peaks of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000120The number of left tunnels of a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000241The number of cyclical small excedances. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000519The largest length of a factor maximising the subword complexity. St000562The number of internal points of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000632The jump number of the poset. St000663The number of right floats of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000873The aix statistic of a permutation. St000922The minimal number such that all substrings of this length are unique. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001665The number of pure excedances of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001372The length of a longest cyclic run of ones of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000014The number of parking functions supported by a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000981The length of the longest zigzag subpath. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001808The box weight or horizontal decoration of a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001885The number of binary words with the same proper border set. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000527The width of the poset. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St001130The number of two successive successions in a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000997The even-odd crank of an integer partition. St000422The energy of a graph, if it is integral. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000116The major index of a semistandard tableau obtained by standardizing. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000736The last entry in the first row of a semistandard tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000958The number of Bruhat factorizations of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001596The number of two-by-two squares inside a skew partition. St001684The reduced word complexity of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001768The number of reduced words of a signed permutation. St001812The biclique partition number of a graph. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000653The last descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. 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. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001114The number of odd descents of a permutation. St001117The game chromatic index of a graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St001964The interval resolution global dimension of a poset. St000135The number of lucky cars of the parking function. St000521The number of distinct subtrees of an ordered tree. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001417The length of a longest palindromic subword of a binary word. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000735The last entry on the main diagonal of a standard tableau.
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