Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000950
(load all 22 compositions to match this statistic)
Mapping
St000950: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2
[1,0,1,0]
 => 2
[1,1,0,0]
 => 5
[1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => 5
[1,1,1,0,0,0]
 => 14
[1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 10
[1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => 10
[1,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => 10
[1,1,1,0,0,0,1,0]
 => 14
[1,1,1,0,0,1,0,0]
 => 14
[1,1,1,0,1,0,0,0]
 => 14
[1,1,1,1,0,0,0,0]
 => 42
[1,0,1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => 10
[1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 8
[1,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,0,0,1,0,0]
 => 10
[1,0,1,1,1,0,1,0,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => 28
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,0,1,1,0,1,0,0]
 => 10
[1,1,0,0,1,1,1,0,0,0]
 => 25
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 10
[1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => 10
[1,1,0,1,1,0,0,1,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => 10
[1,1,0,1,1,1,0,0,0,0]
 => 25
Description
Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.
Matching statistic: St000032
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 5
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 14
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 10
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 10
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 10
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 42
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 10
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 28
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 25
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 25
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000082
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => [.,[.,.]]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => 5
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => 4
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 14
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => 14
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => 10
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => 14
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => 10
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => 10
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => 14
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 42
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => [[[[.,.],.],.],[.,[.,.]]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => [[[[.,.],.],[.,.]],[.,.]]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => [[[[.,.],.],.],[.,[.,.]]]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 14
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => [[[[.,.],[.,.]],.],[.,.]]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => [[[.,.],[.,.]],[.,[.,.]]]
 => 10
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => [[[[.,.],.],[.,.]],[.,.]]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [[[[.,.],.],.],[.,[.,.]]]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 14
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => [[[.,.],[.,[.,.]]],[.,.]]
 => 10
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => [[[.,.],[.,.]],[.,[.,.]]]
 => 10
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 14
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => 42
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,2,6] => [[[[.,[.,.]],.],.],[.,.]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,5,3,1,2,6] => [[[.,[.,.]],.],[.,[.,.]]]
 => 10
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => [[[.,[.,.]],.],[.,[.,.]]]
 => 10
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => 28
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => [[[[.,.],[.,.]],.],[.,.]]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => [[[.,.],[.,.]],[.,[.,.]]]
 => 10
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => [[[[.,.],.],[.,.]],[.,.]]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 14
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => [[[.,.],[.,[.,.]]],[.,.]]
 => 10
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => [[[.,.],[.,.]],[.,[.,.]]]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 14
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => 42
Description
The number of elements smaller than a binary tree in Tamari order.
Matching statistic: St000418
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 5
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 14
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 10
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 10
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 10
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 14
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 42
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 10
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 28
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 25
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 10
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 25
Description
The number of Dyck paths that are weakly below a Dyck path.
Matching statistic: St000421
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 13 = 14 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 9 = 10 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 9 = 10 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 9 = 10 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 13 = 14 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 13 = 14 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 13 = 14 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 41 = 42 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 9 = 10 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 8 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 8 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 9 = 10 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 9 = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 9 = 10 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 27 = 28 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 9 = 10 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 24 = 25 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 9 = 10 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 9 = 10 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 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]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 24 = 25 - 1
Description
The number of Dyck paths that are weakly below a Dyck path, except for the path itself.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 30%
Values
[1,0]
 => [1,1,0,0]
 => [2] => ([],2)
 => 0 = 2 - 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 5 - 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 0 = 2 - 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,14} - 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,5,5,14} - 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,14} - 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,14} - 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {10,10,10,14,14,14,42} - 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6] => ([],6)
 => 0 = 2 - 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000635
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000635: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? ∊ {4,14} - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? ∊ {4,14} - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => ? ∊ {4,4,4,5,5,5,10,10,10,14,14,14,42} - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(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)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? ∊ {2,4,4,4,4,4,4,5,5,5,5,8,8,10,10,10,10,10,10,10,10,10,10,10,10,14,14,14,14,14,14,25,25,28,28,28,28,42,42,42,42,132} - 1
Description
The number of strictly order preserving maps of a poset into itself. A map $f$ is strictly order preserving if $a < b$ implies $f(a) < f(b)$.