Your data matches 210 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000929
(load all 38 compositions to match this statistic)
Mapping
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 0
[1,1,1,1]
 => 1
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 1
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 0
[2,2,2]
 => 0
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 1
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
[5,1,1,1]
 => 0
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000296
(load all 12 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => => ? = 1
[1,1]
 => [1]
 => [1]
 => 10 => 0
[3]
 => []
 => []
 => => ? = 1
[2,1]
 => [1]
 => [1]
 => 10 => 0
[1,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[4]
 => []
 => []
 => => ? = 1
[3,1]
 => [1]
 => [1]
 => 10 => 0
[2,2]
 => [2]
 => [1,1]
 => 110 => 0
[2,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1010 => 0
[5]
 => []
 => []
 => => ? = 1
[4,1]
 => [1]
 => [1]
 => 10 => 0
[3,2]
 => [2]
 => [1,1]
 => 110 => 0
[3,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[2,2,1]
 => [2,1]
 => [3]
 => 1000 => 0
[2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1010 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 10010 => 0
[6]
 => []
 => []
 => => ? = 1
[5,1]
 => [1]
 => [1]
 => 10 => 0
[4,2]
 => [2]
 => [1,1]
 => 110 => 0
[4,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[3,3]
 => [3]
 => [1,1,1]
 => 1110 => 0
[3,2,1]
 => [2,1]
 => [3]
 => 1000 => 0
[3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1010 => 0
[2,2,2]
 => [2,2]
 => [4]
 => 10000 => 0
[2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1100 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 10010 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 10100 => 0
[7]
 => []
 => []
 => => ? = 1
[6,1]
 => [1]
 => [1]
 => 10 => 0
[5,2]
 => [2]
 => [1,1]
 => 110 => 0
[5,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[4,3]
 => [3]
 => [1,1,1]
 => 1110 => 0
[4,2,1]
 => [2,1]
 => [3]
 => 1000 => 0
[4,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1010 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => 10110 => 0
[3,2,2]
 => [2,2]
 => [4]
 => 10000 => 0
[3,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1100 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 10010 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,2,1]
 => 11010 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => 100110 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 10100 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [3,2,1]
 => 101010 => 0
[8]
 => []
 => []
 => => ? = 1
[7,1]
 => [1]
 => [1]
 => 10 => 0
[6,2]
 => [2]
 => [1,1]
 => 110 => 0
[6,1,1]
 => [1,1]
 => [2]
 => 100 => 0
[5,3]
 => [3]
 => [1,1,1]
 => 1110 => 0
[5,2,1]
 => [2,1]
 => [3]
 => 1000 => 0
[5,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1010 => 0
[4,4]
 => [4]
 => [1,1,1,1]
 => 11110 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => 10110 => 0
[4,2,2]
 => [2,2]
 => [4]
 => 10000 => 0
[4,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1100 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 10010 => 0
[3,3,2]
 => [3,2]
 => [5]
 => 100000 => 0
[3,3,1,1]
 => [3,1,1]
 => [4,1]
 => 100010 => 0
[9]
 => []
 => []
 => => ? = 1
[10]
 => []
 => []
 => => ? = 1
[11]
 => []
 => []
 => => ? = 1
[12]
 => []
 => []
 => => ? = 1
Description
The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 33 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => ? => ? => ? = 1
[1,1]
 => [1]
 => 1 => 0 => 0
[3]
 => []
 => ? => ? => ? = 1
[2,1]
 => [1]
 => 1 => 0 => 0
[1,1,1]
 => [1,1]
 => 11 => 01 => 0
[4]
 => []
 => ? => ? => ? = 1
[3,1]
 => [1]
 => 1 => 0 => 0
[2,2]
 => [2]
 => 0 => 1 => 0
[2,1,1]
 => [1,1]
 => 11 => 01 => 0
[1,1,1,1]
 => [1,1,1]
 => 111 => 011 => 0
[5]
 => []
 => ? => ? => ? = 1
[4,1]
 => [1]
 => 1 => 0 => 0
[3,2]
 => [2]
 => 0 => 1 => 0
[3,1,1]
 => [1,1]
 => 11 => 01 => 0
[2,2,1]
 => [2,1]
 => 01 => 11 => 0
[2,1,1,1]
 => [1,1,1]
 => 111 => 011 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0111 => 0
[6]
 => []
 => ? => ? => ? = 1
[5,1]
 => [1]
 => 1 => 0 => 0
[4,2]
 => [2]
 => 0 => 1 => 0
[4,1,1]
 => [1,1]
 => 11 => 01 => 0
[3,3]
 => [3]
 => 1 => 0 => 0
[3,2,1]
 => [2,1]
 => 01 => 11 => 0
[3,1,1,1]
 => [1,1,1]
 => 111 => 011 => 0
[2,2,2]
 => [2,2]
 => 00 => 10 => 0
[2,2,1,1]
 => [2,1,1]
 => 011 => 111 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0111 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 11111 => 01111 => 0
[7]
 => []
 => ? => ? => ? = 1
[6,1]
 => [1]
 => 1 => 0 => 0
[5,2]
 => [2]
 => 0 => 1 => 0
[5,1,1]
 => [1,1]
 => 11 => 01 => 0
[4,3]
 => [3]
 => 1 => 0 => 0
[4,2,1]
 => [2,1]
 => 01 => 11 => 0
[4,1,1,1]
 => [1,1,1]
 => 111 => 011 => 0
[3,3,1]
 => [3,1]
 => 11 => 01 => 0
[3,2,2]
 => [2,2]
 => 00 => 10 => 0
[3,2,1,1]
 => [2,1,1]
 => 011 => 111 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0111 => 0
[2,2,2,1]
 => [2,2,1]
 => 001 => 101 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => 0111 => 1111 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 11111 => 01111 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 111111 => 011111 => 0
[8]
 => []
 => ? => ? => ? = 1
[7,1]
 => [1]
 => 1 => 0 => 0
[6,2]
 => [2]
 => 0 => 1 => 0
[6,1,1]
 => [1,1]
 => 11 => 01 => 0
[5,3]
 => [3]
 => 1 => 0 => 0
[5,2,1]
 => [2,1]
 => 01 => 11 => 0
[5,1,1,1]
 => [1,1,1]
 => 111 => 011 => 0
[4,4]
 => [4]
 => 0 => 1 => 0
[4,3,1]
 => [3,1]
 => 11 => 01 => 0
[4,2,2]
 => [2,2]
 => 00 => 10 => 0
[4,2,1,1]
 => [2,1,1]
 => 011 => 111 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0111 => 0
[3,3,2]
 => [3,2]
 => 10 => 00 => 0
[3,3,1,1]
 => [3,1,1]
 => 111 => 011 => 0
[9]
 => []
 => ? => ? => ? = 1
[10]
 => []
 => ? => ? => ? = 1
[11]
 => []
 => ? => ? => ? = 1
[12]
 => []
 => ? => ? => ? = 1
Description
The defect of a binary word. The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St001696
(load all 4 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => ?
 => ?
 => ? = 1
[1,1]
 => [1]
 => []
 => []
 => 0
[3]
 => []
 => ?
 => ?
 => ? = 1
[2,1]
 => [1]
 => []
 => []
 => 0
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? = 1
[3,1]
 => [1]
 => []
 => []
 => 0
[2,2]
 => [2]
 => []
 => []
 => 0
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? = 1
[4,1]
 => [1]
 => []
 => []
 => 0
[3,2]
 => [2]
 => []
 => []
 => 0
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? = 1
[5,1]
 => [1]
 => []
 => []
 => 0
[4,2]
 => [2]
 => []
 => []
 => 0
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => 0
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? = 1
[6,1]
 => [1]
 => []
 => []
 => 0
[5,2]
 => [2]
 => []
 => []
 => 0
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => 0
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? = 1
[7,1]
 => [1]
 => []
 => []
 => 0
[6,2]
 => [2]
 => []
 => []
 => 0
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => 0
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => 0
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? = 1
[10]
 => []
 => ?
 => ?
 => ? = 1
[11]
 => []
 => ?
 => ?
 => ? = 1
[12]
 => []
 => ?
 => ?
 => ? = 1
Description
The natural major index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000042
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000042: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => []
 => ? = 1
[1,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[3]
 => []
 => []
 => []
 => ? = 1
[2,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[4]
 => []
 => []
 => []
 => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 0
[5]
 => []
 => []
 => []
 => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 0
[6]
 => []
 => []
 => []
 => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => 0
[7]
 => []
 => []
 => []
 => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => 0
[8]
 => []
 => []
 => []
 => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 0
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 0
[9]
 => []
 => []
 => []
 => ? = 1
[10]
 => []
 => []
 => []
 => ? = 1
[11]
 => []
 => []
 => []
 => ? = 1
[12]
 => []
 => []
 => []
 => ? ∊ {0,1}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ?
 => ? ∊ {0,1}
Description
The number of crossings of a perfect matching. This is the number of pairs of edges $((a,b),(c,d))$ such that $a\le c\le b\le d$, i.e., the edges $(a,b)$ and $(c,d)$ cross when drawing the perfect matching as a chord diagram. The generating function for perfect matchings $M$ of $\{1,\dots,2n\}$ according to the number of crossings $\textrm{cr}(M)$ is given by the Touchard-Riordan formula ([2], [4], a bijective proof is given in [7]): $$ \sum_{M} q^{\textrm{cr}(M)} = \frac{1}{(1-q)^n} \sum_{k=0}^n\left(\binom{2n}{n-k}-\binom{2n}{n-k-1}\right)(-1)^k q^{\binom{k+1}{2}} $$
Matching statistic: St000119
(load all 6 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => [] => ? = 1
[1,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[3]
 => []
 => []
 => [] => ? = 1
[2,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[4]
 => []
 => []
 => [] => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[5]
 => []
 => []
 => [] => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[6]
 => []
 => []
 => [] => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[7]
 => []
 => []
 => [] => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 0
[8]
 => []
 => []
 => [] => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 0
[9]
 => []
 => []
 => [] => ? = 1
[10]
 => []
 => []
 => [] => ? = 1
[11]
 => []
 => []
 => [] => ? = 1
[12]
 => []
 => []
 => [] => ? ∊ {0,1}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? ∊ {0,1}
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000232
(load all 3 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => {}
 => ? = 1
[1,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[3]
 => []
 => []
 => {}
 => ? = 1
[2,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[4]
 => []
 => []
 => {}
 => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[5]
 => []
 => []
 => {}
 => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[6]
 => []
 => []
 => {}
 => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 0
[7]
 => []
 => []
 => {}
 => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7}}
 => 0
[8]
 => []
 => []
 => {}
 => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0
[9]
 => []
 => []
 => {}
 => ? = 1
[10]
 => []
 => []
 => {}
 => ? = 1
[11]
 => []
 => []
 => {}
 => ? = 1
[12]
 => []
 => []
 => {}
 => ? ∊ {0,1}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ?
 => ? ∊ {0,1}
Description
The number of crossings of a set partition. This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000297
(load all 9 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => => => ? = 1
[1,1]
 => [1]
 => 10 => 01 => 0
[3]
 => []
 => => => ? = 1
[2,1]
 => [1]
 => 10 => 01 => 0
[1,1,1]
 => [1,1]
 => 110 => 001 => 0
[4]
 => []
 => => => ? = 1
[3,1]
 => [1]
 => 10 => 01 => 0
[2,2]
 => [2]
 => 100 => 011 => 0
[2,1,1]
 => [1,1]
 => 110 => 001 => 0
[1,1,1,1]
 => [1,1,1]
 => 1110 => 0001 => 0
[5]
 => []
 => => => ? = 1
[4,1]
 => [1]
 => 10 => 01 => 0
[3,2]
 => [2]
 => 100 => 011 => 0
[3,1,1]
 => [1,1]
 => 110 => 001 => 0
[2,2,1]
 => [2,1]
 => 1010 => 0101 => 0
[2,1,1,1]
 => [1,1,1]
 => 1110 => 0001 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[6]
 => []
 => => => ? = 1
[5,1]
 => [1]
 => 10 => 01 => 0
[4,2]
 => [2]
 => 100 => 011 => 0
[4,1,1]
 => [1,1]
 => 110 => 001 => 0
[3,3]
 => [3]
 => 1000 => 0111 => 0
[3,2,1]
 => [2,1]
 => 1010 => 0101 => 0
[3,1,1,1]
 => [1,1,1]
 => 1110 => 0001 => 0
[2,2,2]
 => [2,2]
 => 1100 => 0011 => 0
[2,2,1,1]
 => [2,1,1]
 => 10110 => 01001 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 000001 => 0
[7]
 => []
 => => => ? = 1
[6,1]
 => [1]
 => 10 => 01 => 0
[5,2]
 => [2]
 => 100 => 011 => 0
[5,1,1]
 => [1,1]
 => 110 => 001 => 0
[4,3]
 => [3]
 => 1000 => 0111 => 0
[4,2,1]
 => [2,1]
 => 1010 => 0101 => 0
[4,1,1,1]
 => [1,1,1]
 => 1110 => 0001 => 0
[3,3,1]
 => [3,1]
 => 10010 => 01101 => 0
[3,2,2]
 => [2,2]
 => 1100 => 0011 => 0
[3,2,1,1]
 => [2,1,1]
 => 10110 => 01001 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[2,2,2,1]
 => [2,2,1]
 => 11010 => 00101 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 010001 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 000001 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0000001 => 0
[8]
 => []
 => => => ? = 1
[7,1]
 => [1]
 => 10 => 01 => 0
[6,2]
 => [2]
 => 100 => 011 => 0
[6,1,1]
 => [1,1]
 => 110 => 001 => 0
[5,3]
 => [3]
 => 1000 => 0111 => 0
[5,2,1]
 => [2,1]
 => 1010 => 0101 => 0
[5,1,1,1]
 => [1,1,1]
 => 1110 => 0001 => 0
[4,4]
 => [4]
 => 10000 => 01111 => 0
[4,3,1]
 => [3,1]
 => 10010 => 01101 => 0
[4,2,2]
 => [2,2]
 => 1100 => 0011 => 0
[4,2,1,1]
 => [2,1,1]
 => 10110 => 01001 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[3,3,2]
 => [3,2]
 => 10100 => 01011 => 0
[3,3,1,1]
 => [3,1,1]
 => 100110 => 011001 => 0
[9]
 => []
 => => => ? = 1
[10]
 => []
 => => => ? = 1
[11]
 => []
 => => => ? = 1
[12]
 => []
 => => => ? ∊ {0,1}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => 111111111110 => 000000000001 => ? ∊ {0,1}
Description
The number of leading ones in a binary word.
Matching statistic: St000807
(load all 7 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[2]
 => []
 => []
 => [] => ? = 1
[1,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[3]
 => []
 => []
 => [] => ? = 1
[2,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[4]
 => []
 => []
 => [] => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 0
[5]
 => []
 => []
 => [] => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 0
[6]
 => []
 => []
 => [] => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => 0
[7]
 => []
 => []
 => [] => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 0
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => 0
[8]
 => []
 => []
 => [] => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => [1,1] => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 0
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 0
[9]
 => []
 => []
 => [] => ? = 1
[10]
 => []
 => []
 => [] => ? = 1
[11]
 => []
 => []
 => [] => ? = 0
[12]
 => []
 => []
 => [] => ? ∊ {0,0}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? ∊ {0,0}
Description
The sum of the heights of the valleys of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000877
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => => ? = 1
[1,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[3]
 => []
 => []
 => => ? = 1
[2,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[4]
 => []
 => []
 => => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[5]
 => []
 => []
 => => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0
[6]
 => []
 => []
 => => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 0
[7]
 => []
 => []
 => => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => 0
[8]
 => []
 => []
 => => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => 1010 => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
[9]
 => []
 => []
 => => ? = 1
[10]
 => []
 => []
 => => ? = 1
[11]
 => []
 => []
 => => ? = 1
[12]
 => []
 => []
 => => ? ∊ {0,1}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? ∊ {0,1}
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
The following 200 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000974The length of the trunk of an ordered tree. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001394The genus of a permutation. St000769The major index of a composition regarded as a word. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000921The number of internal inversions of a binary word. St000761The number of ascents in an integer composition. St000295The length of the border of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001657The number of twos in an integer partition. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000128The 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000234The number of global ascents of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000358The number of occurrences of the pattern 31-2. 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 $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001513The number of nested exceedences 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. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001381The fertility of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000091The descent variation of a composition. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 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. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000153The number of adjacent cycles of a permutation. St001850The number of Hecke atoms of a permutation. St000658The number of rises of length 2 of a Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000787The number of flips required to make a perfect matching noncrossing. St000326The position of the first one in a binary word after appending a 1 at the end. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001484The number of singletons of an integer partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000546The number of global descents of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. 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. St000478Another weight of a partition according to Alladi. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000390The number of runs of ones in a binary word. St001593This is the number of standard Young tableaux of the given shifted shape. St000214The number of adjacencies of a permutation. St000759The smallest missing part in an integer partition. St001271The competition number of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St000699The toughness times the least common multiple of 1,. St000884The number of isolated descents of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000007The number of saliances of the permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000990The first ascent of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000237The number of small exceedances. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000843The decomposition number of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000989The number of final rises of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000654The first descent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000842The breadth of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001568The smallest positive integer that does not appear twice in the partition. St001826The maximal number of leaves on a vertex of a graph. St001479The number of bridges of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000124The cardinality of the preimage of the Simion-Schmidt map. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000461The rix statistic of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001948The number of augmented double ascents of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000455The second largest eigenvalue of a graph if it is integral.