searching the database
Your data matches 136 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000919
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St000919: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> 0
[[.,.],.]
=> 1
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 1
[[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> 1
[[[.,.],.],.]
=> 1
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 1
[.,[[.,.],[.,.]]]
=> 1
[.,[[.,[.,.]],.]]
=> 1
[.,[[[.,.],.],.]]
=> 1
[[.,.],[.,[.,.]]]
=> 1
[[.,.],[[.,.],.]]
=> 2
[[.,[.,.]],[.,.]]
=> 1
[[[.,.],.],[.,.]]
=> 1
[[.,[.,[.,.]]],.]
=> 1
[[.,[[.,.],.]],.]
=> 2
[[[.,.],[.,.]],.]
=> 1
[[[.,[.,.]],.],.]
=> 1
[[[[.,.],.],.],.]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> 1
[.,[[[[.,.],.],.],.]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> 2
[[.,.],[[[.,.],.],.]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> 1
[[[[.,.],.],.],[.,.]]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> 1
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000659
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000251
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[[.,.],.]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000254
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => {{1,2}}
=> 1
[[.,.],.]
=> [1,2] => {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 1
[[.,.],[.,.]]
=> [3,1,2] => {{1,3},{2}}
=> 1
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => {{1,4},{2,3}}
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => {{1,4},{2,3,5}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => {{1,4},{2,5},{3}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000390
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 1
[.,[[.,.],.]]
=> [2,3,1] => 01 => 1
[[.,.],[.,.]]
=> [3,1,2] => 01 => 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 101 => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 011 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 011 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 011 => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 001 => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 101 => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 001 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 010 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1101 => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1011 => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1001 => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0111 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0111 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0011 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0111 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0011 => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0011 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0101 => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0111 => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0101 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0011 => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0011 => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1011 => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0011 => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1101 => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1001 => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000142
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
Description
The number of even parts of a partition.
Matching statistic: St000157
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 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]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[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
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
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: St000211
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[[.,.],.]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 1
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 1
[[.,.],.]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[[.,.],[.,.]]
=> [3,1,2] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 2
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000291The number of descents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000389The number of runs of ones of odd length in a binary word. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000010The length of the partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000507The number of ascents of a standard tableau. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001354The number of series nodes in the modular decomposition of a graph. St000884The number of isolated descents of a permutation. 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. St000742The number of big ascents of a permutation after prepending zero. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000834The number of right outer peaks of a permutation. St000668The least common multiple of the parts of the partition. St001737The number of descents of type 2 in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity 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. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000386The number of factors DDU in a Dyck path. St000662The staircase size of the code of a permutation. St000306The bounce count of a Dyck path. St000035The number of left outer peaks of a permutation. St000665The number of rafts of a permutation. St000703The number of deficiencies of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000201The number of leaf nodes in a binary tree. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001427The number of descents of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000245The number of ascents of a permutation. St000702The number of weak deficiencies of a permutation. St000647The number of big descents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St000213The number of weak exceedances (also weak excedences) of a permutation. St000619The number of cyclic descents of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St001394The genus of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St001469The holeyness of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000023The number of inner peaks of a permutation. St000317The cycle descent number of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. 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. 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 $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001728The number of invisible descents of a permutation. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000534The number of 2-rises of a permutation. St001597The Frobenius rank of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000392The length of the longest run of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000982The length of the longest constant subword. St000920The logarithmic height of a Dyck path.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!