- FindStat

Your data matches 380 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000386
(load all 73 compositions to match this statistic)

Mapping

St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St001037
(load all 42 compositions to match this statistic)

Mapping

St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000196
(load all 20 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by

$2^{n-2}$ .

Matching statistic: St001840
(load all 15 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a set partition. The Mahonian representation of a set partition

$\{B_1,\dots,B_k\}$ of

$\{1,\dots,n\}$ is the restricted growth word

$w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting

$w_i$ to the index of the block containing

$i$ . The word

$w$ has a descent at position

$i$ if

$w_i > w_{i+1}$ .

Matching statistic: St000201
(load all 17 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size

$n$ , at least

$1$ , with exactly one leaf node for is

$2^{n-1}$ , see [2]. The number of binary tree of size

$n$ , at least

$3$ , with exactly two leaf nodes is

$n(n+1)2^{n-2}$ , see [3].

Matching statistic: St000632
(load all 11 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
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The jump number of the poset. A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St000659
(load all 28 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length at least 2 of a Dyck path.

Matching statistic: St001712
(load all 4 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of natural descents of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.

Matching statistic: St000068
(load all 4 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
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in a poset.

Matching statistic: St000071
(load all 3 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
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of maximal chains in a poset.

The following 370 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000527The width of the poset. St000052The number of valleys of a Dyck path not on the x-axis. St000157The number of descents of a standard tableau. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St000010The length of the partition. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001732The number of peaks visible from the left. St001963The tree-depth of a graph. St000568The hook number of a binary tree. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001469The holeyness of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000619The number of cyclic descents of a permutation. St001175The size of a partition minus the hook length of the base cell. St000142The number of even parts of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000993The multiplicity of the largest part of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001657The number of twos in an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000389The number of runs of ones of odd length in a binary word. St000703The number of deficiencies of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000647The number of big descents of a permutation. St000353The number of inner valleys of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000779The tier of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001092The number of distinct even parts of a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000358The number of occurrences of the pattern 31-2. St001727The number of invisible inversions of a permutation. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000758The length of the longest staircase fitting into an integer composition. St000306The bounce count of a Dyck path. St001394The genus of a permutation. St000023The number of inner peaks of a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant 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

$eA$ . St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001424The number of distinct squares in a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000781The number of proper colouring schemes of a Ferrers diagram. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001728The number of invisible descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000710The number of big deficiencies of a permutation. St000312The number of leaves in a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000711The number of big exceedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000646The number of big ascents of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001587Half of the largest even part of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000015The number of peaks of a Dyck path. St000260The radius of a connected graph. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001568The smallest positive integer that does not appear twice in the partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000897The number of different multiplicities of parts of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001890The maximum magnitude of the Möbius function of a poset. St000251The number of nonsingleton blocks of a set partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000360The number of occurrences of the pattern 32-1. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000523The number of 2-protected nodes of a rooted tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St001307The number of induced stars on four vertices in a graph. St001812The biclique partition number of a graph. St000409The number of pitchforks in a binary tree. St000822The Hadwiger number of the graph. St001471The magnitude of a Dyck path. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001597The Frobenius rank of a skew partition. St000535The rank-width of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001427The number of descents of a signed permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001525The number of symmetric hooks on the diagonal of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001330The hat guessing number of a graph. 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. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001470The cyclic holeyness of a permutation. St000670The reversal length of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001204Call 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000741The Colin de Verdière graph invariant. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000451The length of the longest pattern of the form k 1 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000307The number of rowmotion orbits of a poset. St000352The Elizalde-Pak rank of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000100The number of linear extensions of a poset. St000742The number of big ascents of a permutation after prepending zero. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St000141The maximum drop size of a permutation. St000650The number of 3-rises of a permutation. St000665The number of rafts of a permutation. St001556The number of inversions of the third entry of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St000864The number of circled entries of the shifted recording tableau of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000872The number of very big descents of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001734The lettericity of a graph. St001323The independence gap of a graph. St001642The Prague dimension of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001820The size of the image of the pop stack sorting operator. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000516The number of stretching pairs of a permutation. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000402Half the size of the symmetry class of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000441The number of successions of a permutation. St000731The number of double exceedences of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001549The number of restricted non-inversions between exceedances. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001896The number of right descents of a signed permutations. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001964The interval resolution global dimension of a poset. St000028The number of stack-sorts needed to sort a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001487The number of inner corners of a skew partition. St000308The height of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001520The number of strict 3-descents. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000862The number of parts of the shifted shape of a permutation. St000908The length of the shortest maximal antichain in a poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000181The number of connected components of the Hasse diagram for the poset. St000031The number of cycles in the cycle decomposition of a permutation. 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. St001875The number of simple modules with projective dimension at most 1. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001397Number of pairs of incomparable elements in a finite poset. St000633The size of the automorphism group of a poset. 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. St000058The order of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000630The length of the shortest palindromic decomposition of a binary word. St001435The number of missing boxes in the first row. St000920The logarithmic height of a Dyck path. St000891The number of distinct diagonal sums of a permutation matrix. St000007The number of saliances of the permutation. St000761The number of ascents in an integer composition. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000534The number of 2-rises of 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. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001438The number of missing boxes of a skew partition. St001513The number of nested exceedences of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001871The number of triconnected components of a graph. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001948The number of augmented double ascents 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

$K[x]/(x^n)$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module 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

$eA$ . St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000253The crossing number of a set partition. St000764The number of strong records in an integer composition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000883The number of longest increasing subsequences of a permutation.