- FindStat

Your data matches 116 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000386
(load all 95 compositions to match this statistic)

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St001840
(load all 9 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: St000196
(load all 21 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [.,.]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 0
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 0
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => 0
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 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: St001037
(load all 19 compositions to match this statistic)

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000201
(load all 21 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ 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 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 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: St000157

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000632
(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
Mp00018: Binary trees —left border symmetry⟶ 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),(2,1)],3)
 => 0
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[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),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[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),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[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),(1,2),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[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),(1,2),(2,3)],4)
 => 1
[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),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[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),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[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),(1,2),(2,4),(4,3)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[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),(1,2),(2,4),(4,3)],5)
 => 1
[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),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1

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: St000985
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of positive eigenvalues of the adjacency matrix of the graph.

Matching statistic: St001712
(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
Mp00017: Binary trees —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]
 => [.,[.,.]]
 => [2,1] => [[1],[2]]
 => 0
[1,1,0,0]
 => [[.,.],.]
 => [1,2] => [[1,2]]
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [[1],[2],[3]]
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [[1,2],[3]]
 => 0
[1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => [1,3,2] => [[1,2],[3]]
 => 0
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [[1,3],[2]]
 => 1
[1,1,1,0,0,0]
 => [[[.,.],.],.]
 => [1,2,3] => [[1,2,3]]
 => 0
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[1,2],[3],[4]]
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[1,3],[2],[4]]
 => 1
[1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 0
[1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 0
[1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 1
[1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 1
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 0
[1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[1,2,3],[4],[5]]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,2,3],[4],[5]]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,3],[2,4],[5]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[1,2,4],[3],[5]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[1,2,4],[3],[5]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[1,3,4],[2],[5]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [[1,2,3],[4],[5]]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [[1,2,5],[3],[4]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1

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 3 compositions to match this statistic)

Mapping

Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ 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),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 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),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 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),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 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),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 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),(1,2),(2,3)],4)
 => 2 = 1 + 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),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 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),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 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),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 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),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 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),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1

Description

The number of minimal elements in a poset.

The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000071The number of maximal chains in a poset. St000527The width of the poset. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000919The number of maximal left branches of a binary tree. St000568The hook number of a binary tree. St001083The number of boxed occurrences of 132 in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. 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. St000659The number of rises of length at least 2 of a Dyck path. St000647The number of big descents of a permutation. St000779The tier of a permutation. St000710The number of big deficiencies of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000646The number of big ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000251The number of nonsingleton blocks of a set partition. St000360The number of occurrences of the pattern 32-1. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000021The number of descents of a permutation. St000523The number of 2-protected nodes of a rooted tree. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000325The width of the tree associated to a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001358The largest degree of a regular subgraph 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. St000544The cop number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001354The number of series nodes in the modular decomposition of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001330The hat guessing number of a graph. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001729The number of visible descents of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001896The number of right descents of a signed permutations. St000028The number of stack-sorts needed to sort a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. 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. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000451The length of the longest pattern of the form k 1 2. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000058The order of a permutation. St001487The number of inner corners of a skew partition. St000862The number of parts of the shifted shape of a permutation. St000805The number of peaks of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001948The number of augmented double ascents of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.