- FindStat

Your data matches 287 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000083
(load all 36 compositions to match this statistic)

Mapping

St000083: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left oriented leafs of a binary tree except the first one. In other other words, this is the sum of canopee vector of the tree. The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf. This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.

Matching statistic: St000021
(load all 97 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000024
(load all 110 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000053
(load all 115 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: 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]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 2
[.,[[.,.],.]]
 => [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]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 3
[.,[.,[[.,.],.]]]
 => [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]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 2
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 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]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 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]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[.,.],[.,[[.,.],.]]]
 => [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]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3

Description

The number of valleys of the Dyck path.

Matching statistic: St000168
(load all 20 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of internal nodes of an ordered tree. A node is internal if it is neither the root nor a leaf.

Matching statistic: St000316
(load all 36 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-left-to-right-maxima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a **non-left-to-right-maximum** if there exists a

$j < i$ such that

$\sigma_j > \sigma_i$ .

Matching statistic: St000354
(load all 107 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation

$\pi$ is a value

$i$ such that

$i+1$ appears to the left of

$i$ in

$\pi_1,\pi_2,\dots,\pi_n$ . In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern

$([2,1], {(0,1),(1,1),(2,1)})$ , i.e., the middle row is shaded.

Matching statistic: St000541
(load all 21 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation

$\pi$ of length

$n$ , this is the number of indices

$2 \leq j \leq n$ such that for all

$1 \leq i < j$ , the pair

$(i,j)$ is an inversion of

$\pi$ .

Matching statistic: St000829
(load all 57 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Ulam distance of a permutation to the identity permutation. This is, for a permutation

$\pi$ of

$n$ , given by

$n$ minus the length of the longest increasing subsequence of

$\pi^{-1}$ . In other words, this statistic plus [[St000062]] equals

$n$ .

Matching statistic: St001169
(load all 115 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001169: 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]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 2
[.,[[.,.],.]]
 => [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]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 3
[.,[.,[[.,.],.]]]
 => [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]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 2
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 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]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 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]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[.,.],[.,[[.,.],.]]]
 => [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]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3

Description

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

The following 277 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000167The number of leaves of an ordered tree. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000288The number of ones in a binary word. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001176The size of a partition minus its first part. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000164The number of short pairs. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000912The number of maximal antichains in a poset. St000925The number of topologically connected components of a set partition. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001304The number of maximally independent sets of vertices of a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St000018The number of inversions of a permutation. St000029The depth of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000171The degree of the graph. St000228The size of a partition. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000495The number of inversions of distance at most 2 of a permutation. St000536The pathwidth of a graph. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001777The number of weak descents in an integer composition. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000237The number of small exceedances. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000636The hull number of a graph. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000722The number of different neighbourhoods in a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001622The number of join-irreducible elements of a lattice. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001875The number of simple modules with projective dimension at most 1. St000159The number of distinct parts of the integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000711The number of big exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000619The number of cyclic descents of a permutation. St000242The number of indices that are not cyclical small weak excedances. St001152The number of pairs with even minimum in a perfect matching. St001429The number of negative entries in a signed permutation. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St000236The number of cyclical small weak excedances. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001861The number of Bruhat lower covers of a permutation. St001896The number of right descents of a signed permutations. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St000647The number of big descents of a permutation. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000710The number of big deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice.