- FindStat

Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001486
(load all 4 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St000777

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St000691

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[[]]
 => [.,.]
 => [1] =>  => ? = 1 - 2
[[],[]]
 => [[.,.],.]
 => [1,2] => 1 => 0 = 2 - 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 11 => 0 = 2 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 01 => 1 = 3 - 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 111 => 0 = 2 - 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 101 => 2 = 4 - 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 011 => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 001 => 1 = 3 - 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 001 => 1 = 3 - 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 1111 => 0 = 2 - 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => 1101 => 2 = 4 - 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => 1011 => 2 = 4 - 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => 1001 => 2 = 4 - 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => 1001 => 2 = 4 - 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 0111 => 1 = 3 - 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 0101 => 3 = 5 - 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 0011 => 1 = 3 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 0011 => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 0001 => 1 = 3 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => 0001 => 1 = 3 - 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 0001 => 1 = 3 - 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 0001 => 1 = 3 - 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 0001 => 1 = 3 - 2
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => 11111 => 0 = 2 - 2
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => [1,2,3,5,4,6] => 11101 => 2 = 4 - 2
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => [1,2,4,3,5,6] => 11011 => 2 = 4 - 2
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => [1,2,4,5,3,6] => 11001 => 2 = 4 - 2
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => [1,2,5,4,3,6] => 11001 => 2 = 4 - 2
[[],[[]],[],[],[]]
 => [[[[[.,.],[.,.]],.],.],.]
 => [1,3,2,4,5,6] => 10111 => 2 = 4 - 2
[[],[[]],[[]],[]]
 => [[[[.,.],[.,.]],[.,.]],.]
 => [1,3,2,5,4,6] => 10101 => 4 = 6 - 2
[[],[[],[]],[],[]]
 => [[[[.,.],[[.,.],.]],.],.]
 => [1,3,4,2,5,6] => 10011 => 2 = 4 - 2
[[],[[[]]],[],[]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => [1,4,3,2,5,6] => 10011 => 2 = 4 - 2
[[],[[],[],[]],[]]
 => [[[.,.],[[[.,.],.],.]],.]
 => [1,3,4,5,2,6] => 10001 => 2 = 4 - 2
[[],[[],[[]]],[]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => [1,3,5,4,2,6] => 10001 => 2 = 4 - 2
[[],[[[]],[]],[]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => [1,4,3,5,2,6] => 10001 => 2 = 4 - 2
[[],[[[],[]]],[]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => [1,4,5,3,2,6] => 10001 => 2 = 4 - 2
[[],[[[[]]]],[]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => [1,5,4,3,2,6] => 10001 => 2 = 4 - 2
[[[]],[],[],[],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => [2,1,3,4,5,6] => 01111 => 1 = 3 - 2
[[[]],[],[[]],[]]
 => [[[[.,[.,.]],.],[.,.]],.]
 => [2,1,3,5,4,6] => 01101 => 3 = 5 - 2
[[[]],[[]],[],[]]
 => [[[[.,[.,.]],[.,.]],.],.]
 => [2,1,4,3,5,6] => 01011 => 3 = 5 - 2
[[[]],[[],[]],[]]
 => [[[.,[.,.]],[[.,.],.]],.]
 => [2,1,4,5,3,6] => 01001 => 3 = 5 - 2
[[[]],[[[]]],[]]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => [2,1,5,4,3,6] => 01001 => 3 = 5 - 2
[[[],[]],[],[],[]]
 => [[[[.,[[.,.],.]],.],.],.]
 => [2,3,1,4,5,6] => 00111 => 1 = 3 - 2
[[[[]]],[],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => [3,2,1,4,5,6] => 00111 => 1 = 3 - 2
[[[],[]],[[]],[]]
 => [[[.,[[.,.],.]],[.,.]],.]
 => [2,3,1,5,4,6] => 00101 => 3 = 5 - 2
[[[[]]],[[]],[]]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => [3,2,1,5,4,6] => 00101 => 3 = 5 - 2
[[[],[],[]],[],[]]
 => [[[.,[[[.,.],.],.]],.],.]
 => [2,3,4,1,5,6] => 00011 => 1 = 3 - 2
[[[],[[]]],[],[]]
 => [[[.,[[.,.],[.,.]]],.],.]
 => [2,4,3,1,5,6] => 00011 => 1 = 3 - 2
[[[[]],[]],[],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => [3,2,4,1,5,6] => 00011 => 1 = 3 - 2
[[[[],[]]],[],[]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => [3,4,2,1,5,6] => 00011 => 1 = 3 - 2
[[[[[]]]],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => [4,3,2,1,5,6] => 00011 => 1 = 3 - 2

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St000453

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [1,0]
 => [1] => ([],1)
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(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)
 => 2
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(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
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => ([(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
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [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)
 => 3
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [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)
 => 5
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [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)
 => 5
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [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)
 => 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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)
 => 3
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000455
(load all 3 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 53%●distinct values known / distinct values provided: 29%

Values

[[]]
 => [1,0]
 => [1] => ([],1)
 => ? = 1 - 3
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 2 - 3
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 3 - 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 2 - 3
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(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)
 => -1 = 2 - 3
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(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 - 3
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => ([(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 - 3
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [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)
 => 0 = 3 - 3
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [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)
 => ? = 5 - 3
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [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)
 => ? = 5 - 3
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [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)
 => 0 = 3 - 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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)
 => 0 = 3 - 3
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[[]]]],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[],[[]]],[]]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[[]],[]],[]]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[[],[]]],[]]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[],[[[]]]],[]]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[]],[],[]],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[[]]],[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[],[],[]]],[]]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[],[[]]]],[]]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[[]],[]]],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[[],[]]]],[]]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => -1 = 2 - 3
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[],[]],[],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[[[]]],[],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[[],[]],[[]],[]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[[]]],[[]],[]]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 3
[[],[[],[],[]],[],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[],[[],[[]]],[],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[],[[]]],[],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[[]],[]],[],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[[],[]]],[],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[[[]]]],[],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[],[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
[[[],[],[[]]],[],[]]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001730

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 57%

Values

[[]]
 => [1,0]
 => []
 =>  => ? = 1 - 2
[[],[]]
 => [1,0,1,0]
 => [1]
 => 10 => 0 = 2 - 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1010 => 0 = 2 - 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => 100 => 1 = 3 - 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 0 = 2 - 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 2 = 4 - 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 1 = 3 - 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1 = 3 - 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1 = 3 - 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => 0 = 2 - 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => 2 = 4 - 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => 2 = 4 - 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => 2 = 4 - 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 2 = 4 - 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 1 = 3 - 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 3 = 5 - 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => 1 = 3 - 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => 1 = 3 - 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => 1 = 3 - 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => 1 = 3 - 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => 1 = 3 - 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 100010 => 1 = 3 - 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 10000 => 1 = 3 - 2
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => 1010101010 => ? = 2 - 2
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => 1001101010 => ? = 4 - 2
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => 1010011010 => ? = 4 - 2
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => 1001011010 => ? = 4 - 2
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 1000111010 => ? = 4 - 2
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => 1010100110 => ? = 4 - 2
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 1001100110 => ? = 6 - 2
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => 1010010110 => ? = 4 - 2
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 1010001110 => ? = 4 - 2
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => 1001010110 => ? = 4 - 2
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => 1000110110 => ? = 4 - 2
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => 1001001110 => ? = 4 - 2
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => 1000101110 => ? = 4 - 2
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 1000011110 => ? = 4 - 2
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 101010100 => 1 = 3 - 2
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 100110100 => 3 = 5 - 2
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => 101001100 => 3 = 5 - 2
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => 100101100 => 3 = 5 - 2
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 100011100 => 3 = 5 - 2
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => 101010010 => 1 = 3 - 2
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => 10101000 => 1 = 3 - 2
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => 100110010 => 3 = 5 - 2
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 10011000 => 3 = 5 - 2
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => 101001010 => 1 = 3 - 2
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => 101000110 => 1 = 3 - 2
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => 10100100 => 1 = 3 - 2
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [5,4,1]
 => 10100010 => 1 = 3 - 2
[[[[[]]]],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => 1010000 => 1 = 3 - 2
[[[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => 100101010 => 1 = 3 - 2
[[[],[],[[]]],[]]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => 100011010 => 1 = 3 - 2
[[[],[[]],[]],[]]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => 100100110 => 1 = 3 - 2
[[[],[[],[]]],[]]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1]
 => 100010110 => 1 = 3 - 2
[[[],[[[]]]],[]]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1]
 => 100001110 => 1 = 3 - 2
[[[[]],[],[]],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => 10010100 => 1 = 3 - 2
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => 10001100 => 1 = 3 - 2
[[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [5,3,1]
 => 10010010 => 1 = 3 - 2
[[[[[]]],[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => 1001000 => 1 = 3 - 2
[[[[],[],[]]],[]]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,2,1]
 => 10001010 => 1 = 3 - 2
[[[[],[[]]]],[]]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1,1]
 => 10000110 => 1 = 3 - 2
[[[[[]],[]]],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => 1000100 => 1 = 3 - 2
[[[[[],[]]]],[]]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1]
 => 1000010 => 1 = 3 - 2
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 100000 => 1 = 3 - 2
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => 101010101010 => ? = 2 - 2
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => 100110101010 => ? = 4 - 2
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => 101001101010 => ? = 4 - 2
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => 100101101010 => ? = 4 - 2
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => 100011101010 => ? = 4 - 2
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => 101010011010 => ? = 4 - 2
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => 100110011010 => ? = 6 - 2
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => 101001011010 => ? = 4 - 2
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => 101000111010 => ? = 4 - 2
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => 100101011010 => ? = 4 - 2
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => 100011011010 => ? = 4 - 2
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => 100100111010 => ? = 4 - 2
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => 100010111010 => ? = 4 - 2
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => 100001111010 => ? = 4 - 2
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => 101010100110 => ? = 4 - 2
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => 100110100110 => ? = 6 - 2
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1,1]
 => 101001100110 => ? = 6 - 2
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => 100101100110 => ? = 6 - 2
[[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,1,1]
 => 100011100110 => ? = 6 - 2
[[],[[],[]],[],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => 101010010110 => ? = 4 - 2
[[],[[[]]],[],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1,1,1]
 => 101010001110 => ? = 4 - 2
[[],[[],[]],[[]],[]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1,1]
 => 100110010110 => ? = 6 - 2
[[],[[[]]],[[]],[]]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [6,4,4,1,1,1]
 => 100110001110 => ? = 6 - 2
[[],[[],[],[]],[],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,1]
 => 101001010110 => ? = 4 - 2
[[],[[],[[]]],[],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,1,1]
 => 101000110110 => ? = 4 - 2
[[],[[[]],[]],[],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1,1,1]
 => 101001001110 => ? = 4 - 2
[[],[[[],[]]],[],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1,1,1]
 => 101000101110 => ? = 4 - 2
[[],[[[[]]]],[],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => 101000011110 => ? = 4 - 2
[[],[[],[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,1]
 => 100101010110 => ? = 4 - 2
[[],[[],[],[[]]],[]]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,1,1]
 => 100011010110 => ? = 4 - 2
[[],[[],[[]],[]],[]]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,1,1]
 => 100100110110 => ? = 4 - 2
[[],[[],[[],[]]],[]]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,1,1]
 => 100010110110 => ? = 4 - 2
[[],[[],[[[]]]],[]]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1,1]
 => 100001110110 => ? = 4 - 2
[[],[[[]],[],[]],[]]
 => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1,1,1]
 => 100101001110 => ? = 4 - 2
[[],[[[]],[[]]],[]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,3,1,1,1]
 => 100011001110 => ? = 4 - 2

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Matching statistic: St001488

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.