- FindStat

Your data matches 37 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000108

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => 1
[1,0,1,0]
 => [1]
 => 2
[1,1,0,0]
 => []
 => 1
[1,0,1,0,1,0]
 => [2,1]
 => 5
[1,0,1,1,0,0]
 => [1,1]
 => 3
[1,1,0,0,1,0]
 => [2]
 => 3
[1,1,0,1,0,0]
 => [1]
 => 2
[1,1,1,0,0,0]
 => []
 => 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 14
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 9
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 10
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 7
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 4
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 9
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 6
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 7
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [3]
 => 4
[1,1,1,0,0,1,0,0]
 => [2]
 => 3
[1,1,1,0,1,0,0,0]
 => [1]
 => 2
[1,1,1,1,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 42
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 28
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 32
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 23
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 14
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 32
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 22
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 26
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 19
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 12
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 17
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 13
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 9
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 28
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 19
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 22
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 16
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 10
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 23
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 16
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 19
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 14
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 9
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 13
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 4

Description

The number of partitions contained in the given partition.

Matching statistic: St000420
(load all 6 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 14
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 9
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 10
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 7
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 9
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 7
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 42
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 28
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 32
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 23
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 14
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 32
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 22
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 26
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 19
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 12
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 17
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 13
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 9
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 28
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 19
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 22
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 16
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 10
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 23
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 16
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 19
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 14
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 9
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 13
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4

Description

The number of Dyck paths that are weakly above a Dyck path.

Matching statistic: St000419
(load all 4 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 13 = 14 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 8 = 9 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 9 = 10 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6 = 7 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 8 = 9 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 6 = 7 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 41 = 42 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 27 = 28 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 31 = 32 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 22 = 23 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 13 = 14 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 31 = 32 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 21 = 22 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 25 = 26 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 18 = 19 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 11 = 12 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 16 = 17 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 12 = 13 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 8 = 9 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 27 = 28 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 18 = 19 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 21 = 22 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 15 = 16 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 9 = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 22 = 23 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 15 = 16 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 18 = 19 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 13 = 14 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 8 = 9 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 12 = 13 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 9 = 10 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 4 - 1

Description

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

Matching statistic: St001313
(load all 8 compositions to match this statistic)

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [(1,2)]
 => [2,1] => 1 => 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 101 => 2
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 110 => 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 10101 => 5
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 10110 => 3
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 11001 => 3
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 11010 => 2
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 11100 => 1
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1010101 => 14
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => 1010110 => 9
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => 1011001 => 10
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => 1011010 => 7
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => 1011100 => 4
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => 1100101 => 9
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => 1100110 => 6
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => 1101001 => 7
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => 1101010 => 5
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => 1101100 => 3
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => 1110001 => 4
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => 1110010 => 3
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => 1110100 => 2
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => 1111000 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 101010101 => 42
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => 101010110 => 28
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => 101011001 => 32
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => 101011010 => 23
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => 101011100 => 14
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => 101100101 => 32
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => 101100110 => 22
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => 101101001 => 26
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => 101101010 => 19
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,5,8,4,9,10,7,6,3] => 101101100 => 12
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,6,7,8,5,4,3,10,9] => 101110001 => 17
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => 101110010 => 13
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => 101110100 => 9
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => 101111000 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => 110010101 => 28
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => 110010110 => 19
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => 110011001 => 22
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [3,4,2,1,7,9,6,10,8,5] => 110011010 => 16
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [3,4,2,1,8,9,10,7,6,5] => 110011100 => 10
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => 110100101 => 23
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [3,5,2,6,4,1,9,10,8,7] => 110100110 => 16
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => 110101001 => 19
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => 110101010 => 14
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => 110101100 => 9
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [3,6,2,7,8,5,4,1,10,9] => 110110001 => 13
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => 110110010 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => 110110100 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => 110111000 => 4

Description

The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating

$1$ 's as steps

$(1,0)$ and

$0$ 's as steps

$(0,1)$ . Given a binary word

$w$ , this statistic counts the number of lattice paths from the origin to the same endpoint as

$w$ that stay weakly above

$w$ . See [[St001312]] for this statistic on compositions treated as bounce paths.

Matching statistic: St001389

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 5
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 6
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 9
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 7
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 7
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 9
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 14
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 22
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => 13
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => 16
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 10
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => 19
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => 16
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => 28
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => 23
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => 19
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => 14
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => 12
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => 9
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 10
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => 22
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => 12
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 14
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => 32
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => 26
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => 19

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St000070

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 90% ●values known / values provided: 91%●distinct values known / distinct values provided: 90%

Values

[1,0]
 => []
 => [[],[]]
 => ([],0)
 => ? = 1
[1,0,1,0]
 => [1]
 => [[1],[]]
 => ([],1)
 => 2
[1,1,0,0]
 => []
 => [[],[]]
 => ([],0)
 => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5
[1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 3
[1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 3
[1,1,0,1,0,0]
 => [1]
 => [[1],[]]
 => ([],1)
 => 2
[1,1,1,0,0,0]
 => []
 => [[],[]]
 => ([],0)
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => 14
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 3
[1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 4
[1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 3
[1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => ([],1)
 => 2
[1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ([],0)
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
 => ? ∊ {1,42}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
 => 28
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
 => 32
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => 23
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => 14
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
 => 32
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => 22
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => 26
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => 19
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => 17
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
 => 28
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => 19
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => 22
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => 16
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => 23
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => 16
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => 19
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => 14
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => 14
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => []
 => [[],[]]
 => ([],0)
 => ? ∊ {1,42}

Description

The number of antichains in a poset. An antichain in a poset

$P$ is a subset of elements of

$P$ which are pairwise incomparable. An order ideal is a subset

$I$ of

$P$ such that

$a\in I$ and

$b \leq_P a$ implies

$b \in I$ . Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

Matching statistic: St001232
(load all 28 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 33%●distinct values known / distinct values provided: 24%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001645

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 24%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],1)
 => 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {3,5}
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {3,5}
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,6,7,7,9,9,10,14}
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5

Description

The pebbling number of a connected graph.

Matching statistic: St001330
(load all 7 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 19%●distinct values known / distinct values provided: 14%

Values

[1,0]
 => [2,1] => [1,2] => ([],2)
 => 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [2,3,1] => [1,2,3] => ([],3)
 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
[1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,5}
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,5}
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => ([],4)
 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,4,4,5,6,7,7,9,9,10,14}
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,4,6,2,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,5,2,1,4,6] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,5,6,4,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,6,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [4,2,5,6,1,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,5,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,5,1,3,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [4,6,1,3,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [4,1,3,2,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [5,2,1,4,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [5,1,3,4,2,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [6,2,3,1,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42}
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
 => 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001877
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 19%●distinct values known / distinct values provided: 14%

Values

[1,0]
 => [1] => ([],1)
 => ([(0,1)],2)
 => ? = 1 - 1
[1,0,1,0]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,4,5,6,7,7,9,9,10,14} - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ?
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ?
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ?
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ?
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? ∊ {3,3,4,4,5,5,5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1

Description

Number of indecomposable injective modules with projective dimension 2.

The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000110The number of permutations less than or equal to a permutation in left weak order. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001349The number of different graphs obtained from the given graph by removing an edge. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000358The number of occurrences of the pattern 31-2. St000472The sum of the ascent bottoms of a permutation. St000538The number of even inversions of a permutation. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001311The cyclomatic number of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St000307The number of rowmotion orbits of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.