- FindStat

Your data matches 303 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000920
(load all 51 compositions to match this statistic)

Mapping

St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 2

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St000396
(load all 27 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [[.,.],.]
 => 1
[1,1,0,0]
 => [.,[.,.]]
 => 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 2
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1

Description

The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree

$[[[.,.],[.,.]],[[.,.],[.,.]]]$ : its register function is 3, whereas the dimension of the associated poset is 2.

Matching statistic: St000679
(load all 16 compositions to match this statistic)

Mapping

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [[],[]]
 => 1
[1,1,0,0]
 => [[[]]]
 => 1
[1,0,1,0,1,0]
 => [[],[],[]]
 => 1
[1,0,1,1,0,0]
 => [[],[[]]]
 => 1
[1,1,0,0,1,0]
 => [[[]],[]]
 => 1
[1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[1,1,1,0,0,0]
 => [[[[]]]]
 => 1
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 1
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 1
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 1
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 1
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 1
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 2
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 2
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 1
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 2
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 2
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => 1

Description

The pruning number of an ordered tree. A hanging branch of an ordered tree is a proper factor of the form

$[^r]^r$ for some

$r\geq 1$ . A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch. A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to

$[]$ .

Matching statistic: St001174
(load all 142 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Matching statistic: St000298
(load all 14 compositions to match this statistic)

Mapping

Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

Matching statistic: St000846
(load all 9 compositions to match this statistic)

Mapping

Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of elements covering an element of a poset.

Matching statistic: St000862
(load all 36 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of parts of the shifted shape.

Matching statistic: St001235
(load all 9 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Matching statistic: St001741
(load all 10 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest integer such that all patterns of this size are contained in the permutation.

Matching statistic: St000397
(load all 3 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [[.,.],.]
 => [[[],[]],[]]
 => 2 = 1 + 1
[1,1,0,0]
 => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[],[]],[]],[]],[]],[]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => [[[[[],[]],[]],[[],[]]],[]]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => [[[[[],[]],[[],[]]],[]],[]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => [[[[],[]],[[[],[]],[]]],[]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [[[[],[]],[[],[[],[]]]],[]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [[],[[[[[],[]],[]],[]],[]]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [[],[[[[],[]],[]],[[],[]]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [[],[[[[],[]],[[],[]]],[]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [[],[[[],[]],[[[],[]],[]]]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [[],[[[],[]],[[],[[],[]]]]]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => 2 = 1 + 1

Description

The Strahler number of a rooted tree.

The following 293 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000660The number of rises of length at least 3 of a Dyck path. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001335The cardinality of a minimal cycle-isolating set of a graph. St000058The order of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000308The height of the tree associated to a permutation. St000381The largest part of an integer composition. St000485The length of the longest cycle of a permutation. St000537The cutwidth of a graph. St000544The cop number of a graph. St000701The protection number of a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000845The maximal number of elements covered by an element in a poset. St000903The number of different parts of an integer composition. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001644The dimension of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001792The arboricity of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001962The proper pathwidth of a graph. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000171The degree of the graph. St000271The chromatic index of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000386The number of factors DDU in a Dyck path. St000392The length of the longest run of ones in a binary word. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000779The tier of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001092The number of distinct even parts of a partition. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001393The induced matching number of a graph. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001330The hat guessing number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000098The chromatic number of a graph. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000906The length of the shortest maximal chain in a poset. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001331The size of the minimal feedback vertex set. St001592The maximal number of simple paths between any two different vertices of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000259The diameter of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001597The Frobenius rank of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000662The staircase size of the code of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000640The rank of the largest boolean interval in a poset. St001568The smallest positive integer that does not appear twice in the partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000307The number of rowmotion orbits of a poset. St000822The Hadwiger number of the graph. St001642The Prague dimension of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St000632The jump number of the poset. St001323The independence gap of a graph. St000897The number of different multiplicities of parts of an integer partition. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001060The distinguishing index of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001394The genus of a permutation. St000633The size of the automorphism group of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St000035The number of left outer peaks of a permutation. St000100The number of linear extensions of a poset. St001188The number of simple modules

$S$ with grade

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

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

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

$S$ over the corresponding Nakayama algebra

$A$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001487The number of inner corners of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St001665The number of pure excedances of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001716The 1-improper chromatic number of a graph. St000402Half the size of the symmetry class of a permutation. St000908The length of the shortest maximal antichain in a poset. St001399The distinguishing number of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001820The size of the image of the pop stack sorting operator. St000914The sum of the values of the Möbius function of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000256The number of parts from which one can substract 2 and still get an integer partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001569The maximal modular displacement of a permutation. St000068The number of minimal elements in a poset. St001890The maximum magnitude of the Möbius function of a poset. St001875The number of simple modules with projective dimension at most 1. St001846The number of elements which do not have a complement in the lattice. St001720The minimal length of a chain of small intervals in a lattice. St000031The number of cycles in the cycle decomposition of a permutation. St000741The Colin de Verdière graph invariant. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000768The number of peaks in an integer composition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001728The number of invisible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001896The number of right descents of a signed permutations. St001864The number of excedances of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000807The sum of the heights of the valleys of the associated bargraph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001737The number of descents of type 2 in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000624The normalized sum of the minimal distances to a greater element. St000007The number of saliances of the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St000761The number of ascents in an integer composition. St000021The number of descents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St001469The holeyness of a permutation. St001928The number of non-overlapping descents in a permutation. St001948The number of augmented double ascents of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000664The number of right ropes of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001513The number of nested exceedences of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001549The number of restricted non-inversions between exceedances. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001946The number of descents in a parking function. St000253The crossing number of a set partition. St001488The number of corners of a skew partition. St000883The number of longest increasing subsequences of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000534The number of 2-rises of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation.