Your data matches 330 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001526
(load all 29 compositions to match this statistic)
Mapping
St001526: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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: St001315
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00160: Permutations ā€”graph of inversionsāŸ¶ Graphs
St001315: Graphs āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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 6 compositions to match this statistic)
Mapping
Mp00093: Dyck paths ā€”to binary wordāŸ¶ Binary words
Mp00234: Binary words ā€”valleys-to-peaksāŸ¶ Binary words
St000628: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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 $q$.
Matching statistic: St000836
(load all 32 compositions to match this statistic)
Mapping
Mp00199: Dyck paths ā€”prime Dyck pathāŸ¶ Dyck paths
Mp00031: Dyck paths ā€”to 312-avoiding permutationāŸ¶ Permutations
St000836: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 3 = 4 - 1
Description
The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000837
(load all 30 compositions to match this statistic)
Mapping
Mp00199: Dyck paths ā€”prime Dyck pathāŸ¶ Dyck paths
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
St000837: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 3 = 4 - 1
Description
The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St000381
Mapping
Mp00093: Dyck paths ā€”to binary wordāŸ¶ Binary words
Mp00234: Binary words ā€”valleys-to-peaksāŸ¶ Binary words
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
St000381: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 11 => [1,1,1] => 1
[1,0,1,0]
 => 1010 => 1101 => [1,1,2,1] => 2
[1,1,0,0]
 => 1100 => 1101 => [1,1,2,1] => 2
[1,0,1,0,1,0]
 => 101010 => 110101 => [1,1,2,2,1] => 2
[1,0,1,1,0,0]
 => 101100 => 110101 => [1,1,2,2,1] => 2
[1,1,0,0,1,0]
 => 110010 => 110101 => [1,1,2,2,1] => 2
[1,1,0,1,0,0]
 => 110100 => 111001 => [1,1,1,3,1] => 3
[1,1,1,0,0,0]
 => 111000 => 111001 => [1,1,1,3,1] => 3
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => [1,1,2,1,3,1] => 3
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => [1,1,2,1,3,1] => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => [1,1,1,1,4,1] => 4
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => [1,1,1,1,4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000793
Mapping
Mp00146: Dyck paths ā€”to tunnel matchingāŸ¶ Perfect matchings
Mp00092: Perfect matchings ā€”to set partitionāŸ¶ Set partitions
Mp00115: Set partitions ā€”Kasraoui-ZengāŸ¶ Set partitions
St000793: Set partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => {{1,2}}
 => {{1,2}}
 => 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 2
[1,1,0,0]
 => [(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 2
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2},{3,4},{5,6}}
 => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2},{3,5},{4,6}}
 => 2
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3},{2,4},{5,6}}
 => 2
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5},{4,6}}
 => 3
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2,5},{3,6}}
 => 3
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => {{1,2},{3,4},{5,6},{7,8}}
 => {{1,2},{3,4},{5,6},{7,8}}
 => 2
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => {{1,2},{3,4},{5,8},{6,7}}
 => {{1,2},{3,4},{5,7},{6,8}}
 => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => {{1,2},{3,6},{4,5},{7,8}}
 => {{1,2},{3,5},{4,6},{7,8}}
 => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => {{1,2},{3,8},{4,5},{6,7}}
 => {{1,2},{3,5},{4,7},{6,8}}
 => 3
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => {{1,2},{3,8},{4,7},{5,6}}
 => {{1,2},{3,6},{4,7},{5,8}}
 => 3
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => {{1,4},{2,3},{5,6},{7,8}}
 => {{1,3},{2,4},{5,6},{7,8}}
 => 2
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => {{1,4},{2,3},{5,8},{6,7}}
 => {{1,3},{2,4},{5,7},{6,8}}
 => 2
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => {{1,6},{2,3},{4,5},{7,8}}
 => {{1,3},{2,5},{4,6},{7,8}}
 => 3
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => {{1,8},{2,3},{4,5},{6,7}}
 => {{1,3},{2,5},{4,7},{6,8}}
 => 3
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => {{1,8},{2,3},{4,7},{5,6}}
 => {{1,3},{2,6},{4,7},{5,8}}
 => 3
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => {{1,6},{2,5},{3,4},{7,8}}
 => {{1,4},{2,5},{3,6},{7,8}}
 => 3
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => {{1,8},{2,5},{3,4},{6,7}}
 => {{1,4},{2,5},{3,7},{6,8}}
 => 3
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => {{1,8},{2,7},{3,4},{5,6}}
 => {{1,4},{2,6},{3,7},{5,8}}
 => 4
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => {{1,8},{2,7},{3,6},{4,5}}
 => {{1,5},{2,6},{3,7},{4,8}}
 => 4
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St001119
Mapping
Mp00199: Dyck paths ā€”prime Dyck pathāŸ¶ Dyck paths
Mp00023: Dyck paths ā€”to non-crossing permutationāŸ¶ Permutations
Mp00160: Permutations ā€”graph of inversionsāŸ¶ Graphs
St001119: Graphs āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The length of a shortest maximal path in a graph.
Matching statistic: St001439
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00277: Permutations ā€”catalanizationāŸ¶ Permutations
Mp00066: Permutations ā€”inverseāŸ¶ Permutations
St001439: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 2
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 3
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 3
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 4
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
Description
The number of even weak deficiencies and of odd weak exceedences. For a permutation $\sigma$, this is the number of indices $i$ such that $\sigma(i) \leq i$ if $i$ is even and $\sigma(i) \geq i$ if $i$ is odd. According to [1], $\sigma$ is a '''D-permutation''' if all indices have this property and the coefficients of the characteristic polynomial of the homogenized linial arrangement are given by the number of D-permutations with a given number of cycles.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
Mapping
Mp00093: Dyck paths ā€”to binary wordāŸ¶ Binary words
Mp00234: Binary words ā€”valleys-to-peaksāŸ¶ Binary words
Mp00105: Binary words ā€”complementāŸ¶ Binary words
St000392: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 11 => 00 => 0 = 1 - 1
[1,0,1,0]
 => 1010 => 1101 => 0010 => 1 = 2 - 1
[1,1,0,0]
 => 1100 => 1101 => 0010 => 1 = 2 - 1
[1,0,1,0,1,0]
 => 101010 => 110101 => 001010 => 1 = 2 - 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 001010 => 1 = 2 - 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 001010 => 1 = 2 - 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 000110 => 2 = 3 - 1
[1,1,1,0,0,0]
 => 111000 => 111001 => 000110 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 00100110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 00100110 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 00101010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 00101010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 00011010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 00011010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 00001110 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 00001110 => 3 = 4 - 1
Description
The length of the longest run of ones in a binary word.
The following 320 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000875The semilength of the longest Dyck word in the Catalan factorisation 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. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001280The number of parts of an integer partition that are at least two. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001623The number of doubly irreducible elements of a lattice. St000259The diameter of a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000921The number of internal inversions of a binary word. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001517The length of a longest pair of twins in a permutation. St001691The number of kings in a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000297The number of leading ones in a binary word. 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. 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. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000873The aix statistic of a permutation. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001415The length of the longest palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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. St001118The acyclic chromatic index of a graph. St000264The girth of a graph, which is not a tree. 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 $eA$. 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. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000284The Plancherel distribution on integer partitions. St000667The greatest common divisor of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. 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. 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. St000997The even-odd crank of an integer partition. St001645The pebbling number of a connected graph. 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. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums 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. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. 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. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001948The number of augmented double ascents of a permutation. St000080The rank of the poset. 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. St000245The number of ascents of a permutation. 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. St000624The normalized sum of the minimal distances to a greater element. St000736The last entry in the first row of a semistandard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000989The number of final rises 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. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001684The reduced word complexity of a permutation. St001742The difference of the maximal and the minimal degree in a graph. 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. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001812The biclique partition number of a graph. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St001946The number of descents in a parking function. 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. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000166The depth minus 1 of an ordered tree. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils 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. St000422The energy of a graph, if it is integral. St000443The number of long tunnels of a Dyck path. St000456The monochromatic index of a connected graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. 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. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000809The reduced reflection length of the permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. 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. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001062The maximal size of a block of a set partition. 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 $K[x]/(x^n)$. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001330The hat guessing number of a graph. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. 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. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. 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. St001729The number of visible descents of a permutation. 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. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St000094The depth of an ordered tree. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000521The number of distinct subtrees of an ordered tree. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001417The length of a longest palindromic subword of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000735The last entry on the main diagonal of a standard tableau.