- FindStat

Your data matches 57 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001859
(load all 1454 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
(load all 5 compositions to match this statistic)

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 29 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 6 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 39 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 20 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: St001737
(load all 33 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''.

Matching statistic: St000147

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: 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
[1,2,3] => ([],3)
 => []
 => []
 => 0
[1,3,2] => ([(1,2)],3)
 => [1]
 => [1]
 => 1
[2,1,3] => ([(1,2)],3)
 => [1]
 => [1]
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
[1,2,3,4] => ([],4)
 => []
 => []
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [1]
 => [1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [1]
 => [1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[2,1,3,4] => ([(2,3)],4)
 => [1]
 => [1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => 1
[1,2,3,4,5] => ([],5)
 => []
 => []
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [1]
 => [1]
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [1]
 => [1]
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [1]
 => [1]
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1

Description

The largest part of an integer partition.

The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St001665The number of pure excedances of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000287The number of connected components of a graph. St000286The number of connected components of the complement of a graph. St000456The monochromatic index of a connected graph. St000259The diameter of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001720The minimal length of a chain of small intervals in a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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

$eA$ . St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.