- FindStat

Your data matches 150 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001859
(load all 257 compositions to match this statistic)

Mapping

St001859: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1

Description

The number of factors of the Stanley symmetric function associated with a permutation. For example, the Stanley symmetric function of

$\pi=321645$ equals

$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$

Matching statistic: St000010

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => ([],2)
 => []
 => 0
[2,1] => ([(0,1)],2)
 => [1]
 => 1
[1,2,3] => ([],3)
 => []
 => 0
[1,3,2] => ([(1,2)],3)
 => [1]
 => 1
[2,1,3] => ([(1,2)],3)
 => [1]
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [2]
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [2]
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,2,3,4] => ([],4)
 => []
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2]
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2]
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
[2,1,3,4] => ([(2,3)],4)
 => [1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [1,1]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2]
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2]
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
[1,2,3,4,5] => ([],5)
 => []
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [1]
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [1]
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [1]
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1

Description

The length of the partition.

Matching statistic: St000374
(load all 17 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [1,2] => 0
[2,1] => [1,1,0,0]
 => [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000390
(load all 7 compositions to match this statistic)

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 1 => 0 => 0
[2,1] => 0 => 1 => 1
[1,2,3] => 11 => 00 => 0
[1,3,2] => 10 => 01 => 1
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 11 => 1
[3,1,2] => 00 => 11 => 1
[3,2,1] => 00 => 11 => 1
[1,2,3,4] => 111 => 000 => 0
[1,2,4,3] => 110 => 001 => 1
[1,3,2,4] => 101 => 010 => 1
[1,3,4,2] => 100 => 011 => 1
[1,4,2,3] => 100 => 011 => 1
[1,4,3,2] => 100 => 011 => 1
[2,1,3,4] => 011 => 100 => 1
[2,1,4,3] => 010 => 101 => 2
[2,3,1,4] => 001 => 110 => 1
[2,3,4,1] => 000 => 111 => 1
[2,4,1,3] => 000 => 111 => 1
[2,4,3,1] => 000 => 111 => 1
[3,1,2,4] => 001 => 110 => 1
[3,1,4,2] => 000 => 111 => 1
[3,2,1,4] => 001 => 110 => 1
[3,2,4,1] => 000 => 111 => 1
[3,4,1,2] => 000 => 111 => 1
[3,4,2,1] => 000 => 111 => 1
[4,1,2,3] => 000 => 111 => 1
[4,1,3,2] => 000 => 111 => 1
[4,2,1,3] => 000 => 111 => 1
[4,2,3,1] => 000 => 111 => 1
[4,3,1,2] => 000 => 111 => 1
[4,3,2,1] => 000 => 111 => 1
[1,2,3,4,5] => 1111 => 0000 => 0
[1,2,3,5,4] => 1110 => 0001 => 1
[1,2,4,3,5] => 1101 => 0010 => 1
[1,2,4,5,3] => 1100 => 0011 => 1
[1,2,5,3,4] => 1100 => 0011 => 1
[1,2,5,4,3] => 1100 => 0011 => 1
[1,3,2,4,5] => 1011 => 0100 => 1
[1,3,2,5,4] => 1010 => 0101 => 2
[1,3,4,2,5] => 1001 => 0110 => 1
[1,3,4,5,2] => 1000 => 0111 => 1
[1,3,5,2,4] => 1000 => 0111 => 1
[1,3,5,4,2] => 1000 => 0111 => 1
[1,4,2,3,5] => 1001 => 0110 => 1
[1,4,2,5,3] => 1000 => 0111 => 1
[1,4,3,2,5] => 1001 => 0110 => 1
[1,4,3,5,2] => 1000 => 0111 => 1
[1,4,5,2,3] => 1000 => 0111 => 1
[1,4,5,3,2] => 1000 => 0111 => 1

Description

The number of runs of ones in a binary word.

Matching statistic: St001011
(load all 18 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1

Description

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001280
(load all 4 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => ([],2)
 => [1,1]
 => 0
[2,1] => ([(0,1)],2)
 => [2]
 => 1
[1,2,3] => ([],3)
 => [1,1,1]
 => 0
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => 1
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St001333
(load all 12 compositions to match this statistic)

Mapping

Mp00277: Permutations —catalanization⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,5,3,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1

Description

The cardinality of a minimal edge-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains only the graph with one edge.

Matching statistic: St001665
(load all 11 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 1
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1

Description

The number of pure excedances of a permutation. A pure excedance of a permutation

$\pi$ is a position

$i < \pi_i$ such that there is no

$j < i$ with

$i\leq \pi_j < \pi_i$ .

Matching statistic: St001737
(load all 37 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1

Description

The number of descents of type 2 in a permutation. A position

$i\in[1,n-1]$ is a descent of type 2 of a permutation

$\pi$ of

$n$ letters, if it is a descent and if

$\pi(j) < \pi(i)$ for all

$j < i$ .

Matching statistic: St000035

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

The following 140 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000147The largest part of an integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000291The number of descents of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000659The number of rises of length at least 2 of a Dyck path. St000670The reversal length of a permutation. St000919The number of maximal left branches of a binary tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000668The least common multiple of the parts of the partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000260The radius of a connected graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000618The number of self-evacuating tableaux of given shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001924The number of cells in an integer partition whose arm and leg length coincide. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001568The smallest positive integer that does not appear twice in the partition. St000914The sum of the values of the Möbius function of a poset. St000259The diameter of a connected graph. St000741The Colin de Verdière graph invariant. St001890The maximum magnitude of the Möbius function of a poset. St000775The multiplicity of the largest eigenvalue in a graph. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001330The hat guessing number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000287The number of connected components of a graph. St000286The number of connected components of the complement of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000937The number of positive values of the symmetric group character corresponding to the partition. St000454The largest eigenvalue of a graph if it is integral. St000256The number of parts from which one can substract 2 and still get an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000455The second largest eigenvalue of a graph if it is integral. St001720The minimal length of a chain of small intervals in a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.