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Your data matches 549 different statistics following compositions of up to 3 maps.
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Matching statistic: St001503
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001503: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001503: 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 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[]
=> [1,0]
=> 1 = 0 + 1
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Matching statistic: St000023
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
Mp00149: Permutations āLehmer code rotationā¶ Permutations
St000023: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00149: Permutations āLehmer code rotationā¶ Permutations
St000023: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,3,2,1,5] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [4,3,1,2,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[]
=> [] => [] => 0
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000920
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000920: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000920: 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]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001196
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001196: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001196: 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]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
Description
The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000017
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000017: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000017: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [[1,3],[2]]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [[1,3,4,5,6],[2]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [[1,2,3,4],[5,6]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [[1,2,4,6],[3,5]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [[1,2,3,6],[4,5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [[1,2,3,5,6],[4]]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [[1,2,4,6],[3,5]]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [[1,2,3,6],[4,5]]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [[1,2,3,4,6],[5]]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [[1,2,3,5,6],[4]]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> 0
[]
=> [1,0]
=> [1] => [[1]]
=> 0
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00067: Permutations āFoata bijectionā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
St000021: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00067: Permutations āFoata bijectionā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
St000021: 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
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,2,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,5,3,2] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,2,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[]
=> [] => [] => [] => 0
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000035
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00067: Permutations āFoata bijectionā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
St000035: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00067: Permutations āFoata bijectionā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
St000035: 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
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,2,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,5,3,2] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,2,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[]
=> [] => [] => [] => 0
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
St000039: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
St000039: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,2,1,5,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [5,3,1,4,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [1,6,3,5,2,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,4,2,5,3,6] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [1,6,5,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [1,4,2,6,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [1,4,3,2,5,6] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [1,2,5,3,6,4] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [1,2,5,4,3,6] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [1,5,3,2,6,4] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [1,6,4,2,5,3] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [1,5,4,3,2,6] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [1,2,3,6,5,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [1,2,6,5,4,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [1,6,5,4,3,2] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[]
=> [1,0]
=> [1] => [1] => 0
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000143
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000143: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000143: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [2]
=> 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [3]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [3,1]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [3,1]
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [3,1]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4]
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [4,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [4,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [4,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [4,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [4,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [5,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [3,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [4,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [4,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [4,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [5,1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,3,5,6,4] => [5,1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,2,6,5,3,4] => [4,1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,2,6,4,3,5] => [4,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => [4,2]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,4,6,5] => [5,1]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,3,6,4,5] => [5,1]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,2,6,3,4,5] => [5,1]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [6]
=> 0
[]
=> [1] => [1] => [1]
=> 0
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: St000149
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000149: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00090: Permutations ācycle-as-one-line notationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000149: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,1]
=> 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [3,1]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [2,1,1]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [2,1,1]
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [2,1,1]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [2,1,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [2,1,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [2,1,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [2,1,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [2,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [4,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [3,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [3,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [3,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [2,1,1,1,1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,1,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,3,5,6,4] => [2,1,1,1,1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,2,6,5,3,4] => [3,1,1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,2,6,4,3,5] => [3,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => [2,1,1,1,1]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,4,6,5] => [2,1,1,1,1]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,3,6,4,5] => [2,1,1,1,1]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,2,6,3,4,5] => [2,1,1,1,1]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 0
[]
=> [1] => [1] => [1]
=> 0
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
The following 539 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000150The floored half-sum of the multiplicities of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000662The staircase size of the code of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001175The size of a partition minus the hook length of the base cell. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001728The number of invisible descents of a permutation. St000099The number of valleys of a permutation, including the boundary. St000183The side length of the Durfee square of an integer partition. St000201The number of leaf nodes in a binary tree. St000396The register function (or Horton-Strahler number) of a binary tree. St000527The width of the poset. St000701The protection number of a binary tree. St000897The number of different multiplicities of parts of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000779The tier of a permutation. St001220The width of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000562The number of internal points of a set partition. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001871The number of triconnected components of a graph. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000092The number of outer peaks of a permutation. St000325The width of the tree associated to a permutation. St000354The number of recoils of a permutation. St000381The largest part of an integer composition. St000470The number of runs in a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001432The order dimension of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001716The 1-improper chromatic number of a graph. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000353The number of inner valleys of a permutation. St000539The number of odd inversions of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000829The Ulam distance of a permutation to the identity permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St001330The hat guessing number of a 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 $eA$. St001673The degree of asymmetry of an integer composition. St000390The number of runs of ones in a binary word. St000691The number of changes of a binary word. St001597The Frobenius rank of a skew partition. St000402Half the size of the symmetry class of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001964The interval resolution global dimension of a poset. St000758The length of the longest staircase fitting into an integer composition. St001487The number of inner corners of a skew partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000682The Grundy value of Welter's game on a binary word. St000761The number of ascents in an integer composition. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000805The number of peaks of the associated bargraph. St000903The number of different parts of an integer composition. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001884The number of borders of a binary word. St000253The crossing number of a set partition. St000237The number of small exceedances. St001115The number of even descents of a permutation. St000834The number of right outer peaks of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Mƶbius function of a poset. St000842The breadth of a permutation. St001862The number of crossings of a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000405The number of occurrences of the pattern 1324 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000153The number of adjacent cycles 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. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001866The nesting alignments of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000488The number of cycles of a permutation of length at most 2. St000618The number of self-evacuating tableaux of given shape. St000649The number of 3-excedences of a permutation. St000731The number of double exceedences of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001948The number of augmented double ascents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001586The number of odd parts smaller than the largest even part in an integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001768The number of reduced words of a signed permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001857The number of edges in the reduced word graph of a signed permutation. St001344The neighbouring number of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000241The number of cyclical small excedances. St000284The Plancherel distribution on integer partitions. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000666The number of right tethers of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001411The number of patterns 321 or 3412 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St000078The number of alternating sign matrices whose left key is the permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000477The weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 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. St000567The sum of the products of all pairs of parts. St000570The Edelman-Greene number of a permutation. 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. 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000997The even-odd crank of an integer partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000236The number of cyclical small weak excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000504The cardinality of the first block of a set partition. St000636The hull number of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001062The maximal size of a block of a set partition. St001654The monophonic hull number of a graph. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001561The value of the elementary symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000667The greatest common divisor of the parts of the partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001933The largest multiplicity of a part in an integer partition. St000137The Grundy value of an integer partition. St000260The radius of a connected graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000022The number of fixed points of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000133The "bounce" of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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 $K[x]/(x^n)$. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001524The degree of symmetry of a binary word. St001536The number of cyclic misalignments of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. 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. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000991The number of right-to-left minima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001889The size of the connectivity set of a signed permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000328The maximum number of child nodes in a tree. St000401The size of the symmetry class of a permutation. St000417The size of the automorphism group of the ordered tree. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001075The minimal size of a block of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001569The maximal modular displacement of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000315The number of isolated vertices of a graph. St000894The trace of an alternating sign matrix. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. 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. St001783The number of odd automorphisms of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000096The number of spanning trees of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000450The number of edges minus the number of vertices plus 2 of a graph. St000739The first entry in the last row of a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001204Call 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. St001260The permanent of an alternating sign matrix. St001410The minimal entry of a semistandard tableau. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001651The Frankl number of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001828The Euler characteristic of a graph. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000134The size of the orbit of an alternating sign matrix under gyration. St000259The diameter of a connected graph. St000822The Hadwiger number of the graph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001893The flag descent of a signed permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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