searching the database
Your data matches 9 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: St001050
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 1
[1,2] => {{1},{2}}
=> 2
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => {{1,2,3}}
=> 1
[3,1,2] => {{1,2,3}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 1
[2,4,1,3] => {{1,2,3,4}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> 2
[3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => {{1,2,3,4}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> 1
[4,1,2,3] => {{1,2,3,4}}
=> 1
[4,1,3,2] => {{1,2,4},{3}}
=> 2
[4,2,1,3] => {{1,3,4},{2}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,3,1,2] => {{1,2,3,4}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [2] => 2
[2,1] => [2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3] => 3
[1,3,2] => [1,3,2] => [1,3,2] => [2,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,2] => 2
[2,3,1] => [3,1,2] => [2,3,1] => [2,1] => 1
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1] => 1
[3,2,1] => [2,3,1] => [3,1,2] => [1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,1] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2
[1,3,4,2] => [1,4,2,3] => [1,3,4,2] => [3,1] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1] => 1
[1,4,3,2] => [1,3,4,2] => [1,4,2,3] => [2,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3] => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,2,1] => 1
[2,3,1,4] => [3,1,2,4] => [2,3,1,4] => [2,2] => 2
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => [3,1] => 1
[2,4,1,3] => [4,3,1,2] => [3,4,2,1] => [2,1,1] => 1
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => [2,2] => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 2
[3,1,4,2] => [4,2,1,3] => [3,2,4,1] => [1,2,1] => 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => [1,3] => 3
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => [1,2,1] => 1
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [2,2] => 2
[3,4,2,1] => [4,1,3,2] => [2,4,3,1] => [2,1,1] => 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1
[4,1,3,2] => [3,4,2,1] => [4,3,1,2] => [1,1,2] => 2
[4,2,1,3] => [2,4,3,1] => [4,1,3,2] => [1,2,1] => 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => [1,3] => 3
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => [1,2,1] => 1
[4,3,2,1] => [3,2,4,1] => [4,2,1,3] => [1,1,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [3,2] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => [3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,3] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => [3,2] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => [4,1] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => [3,2] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,4,3,5,2] => [2,2,1] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [2,3] => 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,5,2,4] => [3,2] => 2
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [8,6,4,2,1,3,5,7] => [1,1,1,1,4] => ? = 4
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [4,6,8,2,1,7,5,3] => [3,1,2,1,1] => ? = 1
[7,6,5,8,4,3,2,1] => [8,1,7,2,6,3,5,4] => [2,4,6,8,7,5,3,1] => [4,1,1,1,1] => ? = 1
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [4,2,6,8,7,5,1,3] => ? => ? = 2
[7,6,5,4,8,3,2,1] => [4,8,1,7,2,6,3,5] => [3,5,7,1,8,6,4,2] => [3,2,1,1,1] => ? = 1
[7,4,5,6,8,3,2,1] => [8,1,7,2,4,6,3,5] => [2,4,7,5,8,6,3,1] => ? => ? = 1
[7,6,8,5,3,4,2,1] => [8,1,7,2,6,4,5,3] => [2,4,8,6,7,5,3,1] => [3,2,1,1,1] => ? = 1
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [8,6,2,4,1,3,5,7] => [1,1,2,4] => ? = 4
[7,6,8,4,3,5,2,1] => [4,8,1,7,2,6,5,3] => [3,5,8,1,7,6,4,2] => [3,2,1,1,1] => ? = 1
[7,6,8,3,4,5,2,1] => [8,1,7,2,6,5,4,3] => [2,4,8,7,6,5,3,1] => [3,1,1,1,1,1] => ? = 1
[7,5,4,6,3,8,2,1] => [8,1,7,2,5,3,4,6] => [2,4,6,7,5,8,3,1] => [4,2,1,1] => ? = 1
[5,4,3,6,7,8,2,1] => [3,8,1,5,7,2,4,6] => [3,6,1,7,4,8,5,2] => ? => ? = 1
[5,3,4,6,7,8,2,1] => [8,1,5,7,2,3,4,6] => [2,5,6,7,3,8,4,1] => [4,2,1,1] => ? = 1
[7,5,6,8,4,2,3,1] => ? => ? => ? => ? = 1
[8,7,4,5,6,2,3,1] => [7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ? => ? = 2
[7,8,4,5,6,2,3,1] => [8,1,7,3,4,5,6,2] => [2,8,4,5,6,7,3,1] => [2,4,1,1] => ? = 1
[8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [8,4,2,6,1,3,5,7] => [1,1,2,4] => ? = 4
[8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [8,2,4,6,1,3,5,7] => [1,3,4] => ? = 4
[6,7,8,3,4,2,5,1] => [8,1,6,2,7,5,4,3] => [2,4,8,7,6,3,5,1] => [3,1,1,2,1] => ? = 1
[8,6,7,3,2,4,5,1] => [7,5,2,6,4,3,8,1] => [8,3,6,5,2,4,1,7] => ? => ? = 2
[8,7,2,3,4,5,6,1] => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => ? = 2
[7,8,2,3,4,5,6,1] => [8,1,7,6,5,4,3,2] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => ? = 1
[6,5,7,3,4,2,8,1] => [8,1,6,2,5,4,3,7] => [2,4,7,6,5,3,8,1] => [3,1,1,2,1] => ? = 1
[4,5,6,3,2,7,8,1] => [5,2,8,1,4,3,6,7] => [4,2,6,5,1,7,8,3] => ? => ? = 1
[4,5,3,2,6,7,8,1] => [3,8,1,4,2,5,6,7] => [3,5,1,4,6,7,8,2] => [2,5,1] => ? = 1
[3,4,2,5,6,7,8,1] => [8,1,3,2,4,5,6,7] => [2,4,3,5,6,7,8,1] => [2,5,1] => ? = 1
[7,8,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [6,8,4,2,1,3,5,7] => [2,1,1,4] => ? = 4
[7,6,5,8,4,3,1,2] => [7,1,8,2,6,3,5,4] => [2,4,6,8,7,5,1,3] => [4,1,1,2] => ? = 2
[7,6,4,5,8,3,1,2] => [7,1,8,2,6,3,4,5] => [2,4,6,7,8,5,1,3] => [5,1,2] => ? = 2
[6,7,4,5,8,3,1,2] => [8,2,7,1,6,3,4,5] => [4,2,6,7,8,5,3,1] => [1,4,1,1,1] => ? = 1
[7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [6,8,2,4,1,3,5,7] => [2,2,4] => ? = 4
[7,5,6,8,3,4,1,2] => [7,1,8,2,5,3,6,4] => [2,4,6,8,5,7,1,3] => [4,2,2] => ? = 2
[8,5,4,3,6,7,1,2] => [4,3,8,2,5,6,7,1] => [8,4,2,1,5,6,7,3] => [1,1,1,4,1] => ? = 1
[7,5,4,6,3,8,1,2] => [7,1,8,2,5,3,4,6] => [2,4,6,7,5,8,1,3] => [4,2,2] => ? = 2
[7,3,4,5,6,8,1,2] => [7,1,8,2,3,4,5,6] => [2,4,5,6,7,8,1,3] => [6,2] => ? = 2
[6,4,3,5,7,8,1,2] => [3,8,2,4,5,7,1,6] => [7,3,1,4,5,8,6,2] => [1,1,4,1,1] => ? = 1
[8,7,5,6,4,2,1,3] => [8,3,5,4,6,2,7,1] => [8,6,2,4,3,5,7,1] => ? => ? = 1
[7,8,5,6,4,2,1,3] => [7,1,8,3,5,4,6,2] => [2,8,4,6,5,7,1,3] => ? => ? = 2
[6,7,8,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [4,6,8,2,1,3,5,7] => [3,1,4] => ? = 4
[7,8,5,6,4,1,2,3] => [8,3,5,4,6,1,7,2] => [6,8,2,4,3,5,7,1] => ? => ? = 1
[7,5,6,8,4,1,2,3] => [8,3,6,1,7,2,5,4] => [4,6,2,8,7,3,5,1] => ? => ? = 1
[7,8,4,5,6,1,2,3] => [8,3,4,5,6,1,7,2] => [6,8,2,3,4,5,7,1] => [2,5,1] => ? = 1
[8,6,4,5,7,1,2,3] => [8,3,4,5,7,2,6,1] => [8,6,2,3,4,7,5,1] => ? => ? = 1
[8,7,6,5,3,2,1,4] => [8,4,5,3,6,2,7,1] => [8,6,4,2,3,5,7,1] => [1,1,1,4,1] => ? = 1
[7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [6,4,2,8,1,3,5,7] => [1,1,2,4] => ? = 4
[6,5,7,8,3,2,1,4] => [7,1,6,2,5,3,8,4] => [2,4,6,8,5,3,1,7] => [4,1,1,2] => ? = 2
[6,7,8,5,2,3,1,4] => [8,4,5,2,7,1,6,3] => [6,4,8,2,3,7,5,1] => ? => ? = 1
[7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [6,2,4,8,1,3,5,7] => [1,3,4] => ? = 4
[5,6,7,8,2,3,1,4] => [7,1,5,2,6,3,8,4] => [2,4,6,8,3,5,1,7] => [4,2,2] => ? = 2
[8,7,6,5,3,1,2,4] => [7,2,8,4,5,3,6,1] => [8,2,6,4,5,7,1,3] => ? => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000011
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => ?
=> ?
=> ? = 2
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => ?
=> ?
=> ? = 2
[5,3,4,6,7,8,2,1] => [8,1,5,7,2,3,4,6] => ?
=> ?
=> ? = 1
[7,5,6,8,4,2,3,1] => ? => ?
=> ?
=> ? = 1
[7,8,6,5,3,2,4,1] => [8,1,7,4,5,3,6,2] => ?
=> ?
=> ? = 1
[6,7,5,8,3,2,4,1] => [5,3,8,1,6,2,7,4] => ?
=> ?
=> ? = 1
[8,6,7,5,2,3,4,1] => [7,4,5,2,6,3,8,1] => ?
=> ?
=> ? = 2
[7,8,6,4,3,2,5,1] => [4,8,1,7,5,3,6,2] => ?
=> ?
=> ? = 1
[6,7,8,4,3,2,5,1] => [4,8,1,6,2,7,5,3] => ?
=> ?
=> ? = 1
[7,8,6,3,4,2,5,1] => [8,1,7,5,4,3,6,2] => ?
=> ?
=> ? = 1
[6,7,8,3,4,2,5,1] => [8,1,6,2,7,5,4,3] => ?
=> ?
=> ? = 1
[8,6,7,3,2,4,5,1] => [7,5,2,6,4,3,8,1] => ?
=> ?
=> ? = 2
[6,5,7,3,4,2,8,1] => [8,1,6,2,5,4,3,7] => ?
=> ?
=> ? = 1
[6,5,4,7,2,3,8,1] => [5,2,8,1,6,3,4,7] => ?
=> ?
=> ? = 1
[6,5,7,3,2,4,8,1] => [5,2,8,1,6,4,3,7] => ?
=> ?
=> ? = 1
[5,4,6,3,2,7,8,1] => [8,1,5,2,4,3,6,7] => ?
=> ?
=> ? = 1
[4,5,6,3,2,7,8,1] => [5,2,8,1,4,3,6,7] => ?
=> ?
=> ? = 1
[4,5,3,2,6,7,8,1] => [3,8,1,4,2,5,6,7] => ?
=> ?
=> ? = 1
[8,6,5,7,4,3,1,2] => [8,2,6,3,5,4,7,1] => ?
=> ?
=> ? = 1
[8,5,6,7,3,4,1,2] => [8,2,5,3,6,4,7,1] => ?
=> ?
=> ? = 1
[8,5,4,6,3,7,1,2] => [8,2,5,3,4,6,7,1] => ?
=> ?
=> ? = 1
[6,5,4,7,3,8,1,2] => [8,2,5,3,4,7,1,6] => ?
=> ?
=> ? = 1
[7,4,5,3,6,8,1,2] => [7,1,8,2,4,3,5,6] => ?
=> ?
=> ? = 2
[4,5,6,3,7,8,1,2] => [8,2,5,7,1,4,3,6] => ?
=> ?
=> ? = 1
[6,4,3,5,7,8,1,2] => [3,8,2,4,5,7,1,6] => ?
=> ?
=> ? = 1
[8,7,5,6,4,2,1,3] => [8,3,5,4,6,2,7,1] => ?
=> ?
=> ? = 1
[7,8,5,6,4,1,2,3] => [8,3,5,4,6,1,7,2] => ?
=> ?
=> ? = 1
[8,6,5,7,4,1,2,3] => [8,3,5,4,7,2,6,1] => ?
=> ?
=> ? = 1
[7,5,6,8,4,1,2,3] => [8,3,6,1,7,2,5,4] => ?
=> ?
=> ? = 1
[8,6,4,5,7,1,2,3] => [8,3,4,5,7,2,6,1] => ?
=> ?
=> ? = 1
[6,7,8,5,2,3,1,4] => [8,4,5,2,7,1,6,3] => ?
=> ?
=> ? = 1
[6,7,5,8,2,3,1,4] => [7,1,6,3,5,2,8,4] => ?
=> ?
=> ? = 2
[7,5,6,8,3,1,2,4] => [7,2,5,3,6,1,8,4] => ?
=> ?
=> ? = 2
[8,5,6,7,2,1,3,4] => [5,2,8,4,7,3,6,1] => ?
=> ?
=> ? = 1
[6,7,8,5,1,2,3,4] => [8,4,5,1,6,2,7,3] => ?
=> ?
=> ? = 1
[8,6,7,3,2,4,1,5] => [8,5,2,6,4,3,7,1] => ?
=> ?
=> ? = 1
[8,7,6,2,3,4,1,5] => [8,5,3,6,4,2,7,1] => ?
=> ?
=> ? = 1
[7,8,6,3,4,1,2,5] => [8,5,4,3,6,1,7,2] => ?
=> ?
=> ? = 1
[7,8,6,4,1,2,3,5] => [4,8,5,1,7,3,6,2] => ?
=> ?
=> ? = 1
[6,7,8,4,1,2,3,5] => [4,8,5,1,6,2,7,3] => ?
=> ?
=> ? = 1
[8,7,6,3,2,1,4,5] => [8,5,2,7,4,3,6,1] => ?
=> ?
=> ? = 1
[7,6,8,3,2,1,4,5] => [8,5,2,6,1,7,4,3] => ?
=> ?
=> ? = 1
[6,7,8,3,1,2,4,5] => [8,5,1,6,2,7,4,3] => ?
=> ?
=> ? = 1
[7,8,5,3,2,1,4,6] => [8,6,1,7,4,3,5,2] => ?
=> ?
=> ? = 1
[8,7,5,3,1,2,4,6] => [8,6,2,7,4,3,5,1] => ?
=> ?
=> ? = 1
[7,8,4,3,2,1,5,6] => [4,3,8,6,1,7,5,2] => ?
=> ?
=> ? = 1
[7,8,4,2,3,1,5,6] => [8,6,1,7,5,3,4,2] => ?
=> ?
=> ? = 1
[7,8,3,2,4,1,5,6] => [3,8,6,1,7,5,4,2] => ?
=> ?
=> ? = 1
[8,7,4,2,1,3,5,6] => [8,6,3,4,2,7,5,1] => ?
=> ?
=> ? = 1
[7,8,4,1,2,3,5,6] => [8,6,3,4,1,7,5,2] => ?
=> ?
=> ? = 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000745
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> 1
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 2
[2,1] => [2,1] => [1,2] => [[1,2]]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,3,2] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [2,3,1] => [[1,3],[2]]
=> 2
[2,3,1] => [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [3,2,1] => [1,2,3] => [[1,2,3]]
=> 1
[3,2,1] => [2,3,1] => [2,1,3] => [[1,3],[2]]
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,3,1,4] => [3,1,2,4] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[2,4,1,3] => [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => [[1,3,4],[2]]
=> 2
[3,1,4,2] => [4,2,1,3] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => [[1,2],[3,4]]
=> 1
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,1,3,2] => [3,4,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 2
[4,2,1,3] => [2,4,3,1] => [3,1,2,4] => [[1,2,4],[3]]
=> 1
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,3,2,1] => [3,2,4,1] => [2,3,1,4] => [[1,3,4],[2]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => [[1,2],[3],[4],[5]]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => [[1,2],[3],[4],[5]]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => [[1,2,3],[4],[5]]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => [[1,2,4],[3],[5]]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => [[1,2],[3,4],[5]]
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => [[1,3],[2,4],[5]]
=> 2
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [2,7,1,8,3,6,4,5] => ?
=> ? = 2
[7,5,6,8,4,3,2,1] => [6,3,8,1,7,2,5,4] => [3,6,1,8,2,7,4,5] => ?
=> ? = 1
[6,5,7,4,8,3,2,1] => [4,8,1,6,3,7,2,5] => [5,1,8,3,6,2,7,4] => ?
=> ? = 1
[6,7,4,5,8,3,2,1] => [7,2,8,1,6,3,4,5] => [2,7,1,8,3,6,5,4] => ?
=> ? = 2
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [2,7,3,5,4,6,1,8] => ?
=> ? = 2
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => [2,7,4,6,3,5,1,8] => ?
=> ? = 2
[7,6,5,8,3,4,2,1] => [5,3,8,1,7,2,6,4] => [4,6,1,8,2,7,3,5] => ?
=> ? = 1
[5,4,3,6,7,8,2,1] => [3,8,1,5,7,2,4,6] => [6,1,8,4,2,7,5,3] => ?
=> ? = 1
[7,8,6,5,4,2,3,1] => [5,4,8,1,7,3,6,2] => [4,5,1,8,2,6,3,7] => ?
=> ? = 1
[8,7,5,6,4,2,3,1] => [7,3,5,4,6,2,8,1] => [2,6,4,5,3,7,1,8] => ?
=> ? = 2
[7,6,5,8,4,2,3,1] => [6,2,8,1,7,3,5,4] => [3,7,1,8,2,6,4,5] => ?
=> ? = 1
[7,5,6,8,4,2,3,1] => ? => ? => ?
=> ? = 1
[8,7,4,5,6,2,3,1] => [7,3,4,5,6,2,8,1] => [2,6,5,4,3,7,1,8] => ?
=> ? = 2
[7,8,4,5,6,2,3,1] => [8,1,7,3,4,5,6,2] => [1,8,2,6,5,4,3,7] => ?
=> ? = 1
[7,8,6,5,3,2,4,1] => [8,1,7,4,5,3,6,2] => [1,8,2,5,4,6,3,7] => ?
=> ? = 1
[8,7,5,6,2,3,4,1] => [7,4,6,3,5,2,8,1] => [2,5,3,6,4,7,1,8] => ?
=> ? = 2
[7,8,6,4,3,2,5,1] => [4,8,1,7,5,3,6,2] => [5,1,8,2,4,6,3,7] => ?
=> ? = 1
[8,7,6,3,4,2,5,1] => [7,5,4,3,6,2,8,1] => [2,4,5,6,3,7,1,8] => ?
=> ? = 2
[7,8,6,3,4,2,5,1] => [8,1,7,5,4,3,6,2] => [1,8,2,4,5,6,3,7] => ?
=> ? = 1
[7,6,8,4,2,3,5,1] => [4,8,1,7,5,2,6,3] => [5,1,8,2,4,7,3,6] => ?
=> ? = 1
[7,8,6,2,3,4,5,1] => [8,1,7,5,3,6,4,2] => [1,8,2,4,6,3,5,7] => ?
=> ? = 1
[6,5,4,7,3,2,8,1] => [8,1,6,2,5,3,4,7] => [1,8,3,7,4,6,5,2] => ?
=> ? = 1
[6,5,4,3,7,2,8,1] => [4,3,8,1,6,2,5,7] => [5,6,1,8,3,7,4,2] => ?
=> ? = 1
[4,5,6,3,7,2,8,1] => [8,1,4,3,6,2,5,7] => [1,8,5,6,3,7,4,2] => ?
=> ? = 1
[6,4,3,5,7,2,8,1] => [3,8,1,6,2,4,5,7] => [6,1,8,3,7,5,4,2] => ?
=> ? = 1
[4,3,5,6,7,2,8,1] => [8,1,4,6,2,3,5,7] => [1,8,5,3,7,6,4,2] => ?
=> ? = 1
[6,5,4,7,2,3,8,1] => [5,2,8,1,6,3,4,7] => [4,7,1,8,3,6,5,2] => ?
=> ? = 1
[5,4,6,7,2,3,8,1] => [6,3,8,1,5,2,4,7] => [3,6,1,8,4,7,5,2] => ?
=> ? = 1
[6,5,7,3,2,4,8,1] => [5,2,8,1,6,4,3,7] => [4,7,1,8,3,5,6,2] => ?
=> ? = 1
[6,7,4,3,2,5,8,1] => [4,3,8,1,6,5,2,7] => [5,6,1,8,3,4,7,2] => ?
=> ? = 1
[5,4,6,3,2,7,8,1] => [8,1,5,2,4,3,6,7] => [1,8,4,7,5,6,3,2] => ?
=> ? = 1
[4,5,6,3,2,7,8,1] => [5,2,8,1,4,3,6,7] => [4,7,1,8,5,6,3,2] => ?
=> ? = 1
[5,4,6,2,3,7,8,1] => [4,2,8,1,5,3,6,7] => [5,7,1,8,4,6,3,2] => ?
=> ? = 1
[4,3,5,2,6,7,8,1] => [8,1,4,2,3,5,6,7] => [1,8,5,7,6,4,3,2] => ?
=> ? = 1
[3,4,5,2,6,7,8,1] => [4,2,8,1,3,5,6,7] => [5,7,1,8,6,4,3,2] => ?
=> ? = 1
[8,6,7,5,4,3,1,2] => [5,4,8,2,6,3,7,1] => [4,5,1,7,3,6,2,8] => ?
=> ? = 1
[8,6,5,7,4,3,1,2] => [8,2,6,3,5,4,7,1] => [1,7,3,6,4,5,2,8] => ?
=> ? = 1
[7,6,4,5,8,3,1,2] => [7,1,8,2,6,3,4,5] => [2,8,1,7,3,6,5,4] => ?
=> ? = 2
[8,6,5,7,3,4,1,2] => [5,3,8,2,6,4,7,1] => [4,6,1,7,3,5,2,8] => ?
=> ? = 1
[8,4,3,5,6,7,1,2] => [3,8,2,4,5,6,7,1] => [6,1,7,5,4,3,2,8] => ?
=> ? = 1
[7,5,4,6,3,8,1,2] => [7,1,8,2,5,3,4,6] => [2,8,1,7,4,6,5,3] => ?
=> ? = 2
[6,5,4,7,3,8,1,2] => [8,2,5,3,4,7,1,6] => [1,7,4,6,5,2,8,3] => ?
=> ? = 1
[7,4,5,3,6,8,1,2] => [7,1,8,2,4,3,5,6] => [2,8,1,7,5,6,4,3] => ?
=> ? = 2
[8,7,6,5,4,2,1,3] => [5,4,8,3,6,2,7,1] => [4,5,1,6,3,7,2,8] => ?
=> ? = 1
[8,7,5,6,4,2,1,3] => [8,3,5,4,6,2,7,1] => [1,6,4,5,3,7,2,8] => ?
=> ? = 1
[7,8,5,6,4,2,1,3] => [7,1,8,3,5,4,6,2] => [2,8,1,6,4,5,3,7] => ?
=> ? = 2
[7,5,6,8,4,2,1,3] => [7,1,8,3,6,2,5,4] => [2,8,1,6,3,7,4,5] => ?
=> ? = 2
[5,4,6,7,8,2,1,3] => [8,3,6,2,4,7,1,5] => [1,6,3,7,5,2,8,4] => ?
=> ? = 1
[7,8,5,6,4,1,2,3] => [8,3,5,4,6,1,7,2] => [1,6,4,5,3,8,2,7] => ?
=> ? = 1
[7,5,6,8,4,1,2,3] => [8,3,6,1,7,2,5,4] => [1,6,3,8,2,7,4,5] => ?
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000678
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[2,3,1,4] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,3,1] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[3,2,4,1] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,1,2] => [3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,3,1,2] => [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [5,4,8,1,7,2,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[8,6,5,7,4,3,2,1] => [7,2,6,3,5,4,8,1] => [7,2,6,3,5,4,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[7,6,5,8,4,3,2,1] => [8,1,7,2,6,3,5,4] => [8,1,7,2,6,3,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [7,2,8,1,6,3,5,4] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 2
[7,5,6,8,4,3,2,1] => [6,3,8,1,7,2,5,4] => [6,3,8,1,7,2,5,4] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1
[6,5,7,8,4,3,2,1] => [8,1,6,3,7,2,5,4] => [8,1,6,3,7,2,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,4,5,7,3,2,1] => [7,2,6,3,4,5,8,1] => [2,3,7,4,6,5,8,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => [7,4,2,3,6,5,8,1] => ?
=> ? = 2
[7,6,5,4,8,3,2,1] => [4,8,1,7,2,6,3,5] => [4,1,8,2,7,6,3,5] => ?
=> ? = 1
[6,5,7,4,8,3,2,1] => [4,8,1,6,3,7,2,5] => [1,4,6,8,7,3,2,5] => ?
=> ? = 1
[6,7,4,5,8,3,2,1] => [7,2,8,1,6,3,4,5] => [7,2,1,3,8,4,6,5] => ?
=> ? = 2
[7,4,5,6,8,3,2,1] => [8,1,7,2,4,6,3,5] => [8,7,1,2,4,3,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,5,6,7,8,3,2,1] => [6,3,8,1,4,7,2,5] => [3,6,8,1,4,2,7,5] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [2,7,6,4,5,3,8,1] => [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [3,5,6,4,7,2,8,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => [2,3,7,5,6,4,8,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[7,6,5,8,3,4,2,1] => [5,3,8,1,7,2,6,4] => [3,5,8,1,7,2,6,4] => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[7,6,8,4,3,5,2,1] => [4,8,1,7,2,6,5,3] => [8,4,7,1,6,2,5,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,3,4,5,2,1] => [7,2,6,5,4,3,8,1] => [7,6,5,2,4,3,8,1] => ?
=> ? = 2
[7,6,8,3,4,5,2,1] => [8,1,7,2,6,5,4,3] => [8,7,6,1,5,2,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,3,4,5,2,1] => [7,2,8,1,6,5,4,3] => [7,8,2,6,5,1,4,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[8,3,4,5,6,7,2,1] => [7,2,3,4,5,6,8,1] => [2,3,4,5,7,6,8,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[5,6,7,4,3,8,2,1] => [4,8,1,5,3,7,2,6] => [1,4,8,5,7,3,2,6] => ?
=> ? = 1
[7,5,4,6,3,8,2,1] => [8,1,7,2,5,3,4,6] => [1,8,2,3,5,4,7,6] => ?
=> ? = 1
[7,4,5,6,3,8,2,1] => [5,3,8,1,7,2,4,6] => [5,1,8,3,2,7,4,6] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[7,5,4,3,6,8,2,1] => [4,3,8,1,7,2,5,6] => [4,1,8,7,3,2,5,6] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,4,3,6,7,8,2,1] => [3,8,1,5,7,2,4,6] => [8,3,1,7,2,5,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,3,4,6,7,8,2,1] => [8,1,5,7,2,3,4,6] => [1,2,8,5,3,4,7,6] => ?
=> ? = 1
[3,4,5,6,7,8,2,1] => [8,1,3,5,7,2,4,6] => [8,5,3,1,2,4,7,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,6,5,4,2,3,1] => [5,4,8,1,7,3,6,2] => [5,8,4,1,7,3,6,2] => ?
=> ? = 1
[8,6,7,5,4,2,3,1] => [5,4,6,2,7,3,8,1] => [5,4,2,6,7,3,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 4
[8,7,5,6,4,2,3,1] => [7,3,5,4,6,2,8,1] => [5,3,4,7,6,2,8,1] => ?
=> ? = 2
[7,6,5,8,4,2,3,1] => [6,2,8,1,7,3,5,4] => [2,6,8,1,7,3,5,4] => ?
=> ? = 1
[7,5,6,8,4,2,3,1] => ? => ? => ?
=> ? = 1
[8,7,4,5,6,2,3,1] => [7,3,4,5,6,2,8,1] => [3,4,5,7,6,2,8,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,8,4,5,6,2,3,1] => [8,1,7,3,4,5,6,2] => [1,3,4,8,7,5,6,2] => ?
=> ? = 1
[7,5,6,4,8,2,3,1] => [4,8,1,7,3,6,2,5] => [8,1,4,7,6,3,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,5,6,7,8,2,3,1] => [8,1,4,7,3,6,2,5] => [4,8,7,3,1,2,6,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,3,2,4,1] => [7,4,5,3,6,2,8,1] => [4,5,3,7,6,2,8,1] => [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,8,6,5,3,2,4,1] => [8,1,7,4,5,3,6,2] => ? => ?
=> ? = 1
[7,6,8,5,3,2,4,1] => [6,2,8,1,7,4,5,3] => [8,2,6,1,7,4,5,3] => ?
=> ? = 1
[6,7,8,5,3,2,4,1] => [8,1,6,2,7,4,5,3] => [8,1,6,2,7,4,5,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [3,5,2,6,7,4,8,1] => [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 4
[6,7,5,8,3,2,4,1] => [5,3,8,1,6,2,7,4] => [8,3,1,5,2,6,7,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,8,3,2,4,1] => [6,2,8,1,5,3,7,4] => [6,2,1,8,3,5,7,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 1
[8,6,7,5,2,3,4,1] => [7,4,5,2,6,3,8,1] => [4,2,5,7,6,3,8,1] => [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,8,5,2,3,4,1] => [8,1,7,4,5,2,6,3] => [4,1,2,8,7,5,6,3] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8,7,5,6,2,3,4,1] => [7,4,6,3,5,2,8,1] => [7,4,3,6,5,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000989
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,2,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [3,4,2,1] => [1,4,3,2] => 0 = 1 - 1
[1,4,3,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[2,4,3,1] => [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[3,1,4,2] => [4,2,1,3] => [3,1,4,2] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[3,2,4,1] => [2,4,1,3] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,4,2] => [3,4,1,2] => [2,4,1,3] => 1 = 2 - 1
[3,4,2,1] => [4,1,3,2] => [2,4,3,1] => [1,4,3,2] => 0 = 1 - 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,1,3,2] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
[4,2,1,3] => [2,4,3,1] => [4,2,3,1] => [4,1,3,2] => 0 = 1 - 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[4,3,1,2] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[4,3,2,1] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,3,5,2,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [3,1,5,4,2] => [2,1,5,4,3] => 0 = 1 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,4,2,1,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,5,3,2,4] => [3,4,1,5,2] => [1,3,2,5,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => 2 = 3 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [2,1,4,5,3] => [2,1,3,5,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1 = 2 - 1
[1,4,5,3,2] => [1,5,2,4,3] => [3,2,5,4,1] => [2,1,5,4,3] => 0 = 1 - 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [3,4,1,2,7,6,5] => [2,4,1,3,7,6,5] => ? = 1 - 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [3,4,1,7,6,2,5] => [2,4,1,7,6,3,5] => ? = 2 - 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [3,4,1,7,2,6,5] => [2,4,1,7,3,6,5] => ? = 1 - 1
[1,2,7,3,5,6,4] => [1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => [2,4,7,6,1,3,5] => ? = 3 - 1
[1,2,7,3,6,5,4] => [1,2,6,5,7,4,3] => [4,5,7,6,2,1,3] => [1,4,7,6,3,2,5] => ? = 2 - 1
[1,2,7,4,3,6,5] => [1,2,4,6,7,5,3] => [3,4,7,2,6,1,5] => [1,4,7,3,6,2,5] => ? = 2 - 1
[1,2,7,4,6,3,5] => [1,2,4,7,5,6,3] => [3,4,7,2,1,6,5] => [1,4,7,3,2,6,5] => ? = 1 - 1
[1,2,7,5,3,6,4] => [1,2,6,7,4,5,3] => [3,4,7,1,6,2,5] => [2,4,7,1,6,3,5] => ? = 2 - 1
[1,2,7,5,4,3,6] => [1,2,5,4,7,6,3] => [4,5,7,3,2,6,1] => [1,4,7,3,2,6,5] => ? = 1 - 1
[1,2,7,5,4,6,3] => [1,2,5,4,6,7,3] => [3,4,7,2,1,5,6] => [1,4,7,3,2,5,6] => ? = 3 - 1
[1,2,7,5,6,3,4] => [1,2,7,4,5,6,3] => [3,4,7,1,2,6,5] => [2,4,7,1,3,6,5] => ? = 1 - 1
[1,2,7,6,3,4,5] => [1,2,6,4,7,5,3] => [4,5,7,3,6,1,2] => [1,4,7,3,6,2,5] => ? = 2 - 1
[1,2,7,6,4,3,5] => [1,2,7,5,4,6,3] => [4,5,7,3,1,6,2] => [1,4,7,3,2,6,5] => ? = 1 - 1
[1,2,7,6,4,5,3] => [1,2,6,5,4,7,3] => [4,5,7,3,2,1,6] => [1,5,7,4,3,2,6] => ? = 2 - 1
[1,3,2,7,6,5,4] => [1,3,2,6,5,7,4] => [4,5,3,7,2,1,6] => [1,5,4,7,3,2,6] => ? = 2 - 1
[1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 4 - 1
[1,3,4,2,5,7,6] => [1,4,2,3,5,7,6] => [3,2,5,4,6,7,1] => [2,1,4,3,5,7,6] => ? = 1 - 1
[1,3,4,2,6,5,7] => [1,4,2,3,6,5,7] => [3,2,5,4,6,1,7] => [2,1,4,3,6,5,7] => ? = 2 - 1
[1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => [3,2,5,4,1,7,6] => [2,1,5,4,3,7,6] => ? = 1 - 1
[1,3,4,2,7,5,6] => [1,4,2,3,7,6,5] => [4,3,6,5,7,2,1] => [2,1,4,3,7,6,5] => ? = 1 - 1
[1,3,4,2,7,6,5] => [1,4,2,3,6,7,5] => [3,2,5,4,7,1,6] => [2,1,5,4,7,3,6] => ? = 2 - 1
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 3 - 1
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [3,2,4,6,5,7,1] => [2,1,3,5,4,7,6] => ? = 1 - 1
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
[1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 1 - 1
[1,3,4,5,7,2,6] => [1,7,6,2,3,4,5] => [3,1,2,4,7,6,5] => [3,1,2,4,7,6,5] => ? = 1 - 1
[1,3,4,5,7,6,2] => [1,6,7,2,3,4,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 2 - 1
[1,3,4,6,2,5,7] => [1,6,5,2,3,4,7] => [3,1,2,6,5,4,7] => [3,1,2,6,5,4,7] => ? = 2 - 1
[1,3,4,6,2,7,5] => [1,7,5,2,3,4,6] => [3,1,2,6,4,7,5] => [3,1,2,5,4,7,6] => ? = 1 - 1
[1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => [2,1,3,6,4,5,7] => [2,1,3,6,4,5,7] => ? = 3 - 1
[1,3,4,6,5,7,2] => [1,5,7,2,3,4,6] => [2,1,3,4,6,7,5] => [2,1,3,4,5,7,6] => ? = 1 - 1
[1,3,4,6,7,2,5] => [1,6,2,3,4,7,5] => [3,2,4,6,7,1,5] => [2,1,3,5,7,4,6] => ? = 2 - 1
[1,3,4,6,7,5,2] => [1,7,2,3,4,6,5] => [3,2,4,5,7,6,1] => [2,1,3,4,7,6,5] => ? = 1 - 1
[1,3,4,7,2,5,6] => [1,7,6,5,2,3,4] => [4,1,2,7,6,5,3] => [3,1,2,7,6,5,4] => ? = 1 - 1
[1,3,4,7,2,6,5] => [1,6,7,5,2,3,4] => [3,1,2,7,6,4,5] => [3,1,2,7,6,4,5] => ? = 2 - 1
[1,3,4,7,5,2,6] => [1,5,7,6,2,3,4] => [3,1,2,7,5,6,4] => [3,1,2,7,4,6,5] => ? = 1 - 1
[1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 3 - 1
[1,3,4,7,6,2,5] => [1,7,5,6,2,3,4] => [3,1,2,7,4,6,5] => [3,1,2,7,4,6,5] => ? = 1 - 1
[1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => [3,1,2,7,5,4,6] => [3,1,2,7,5,4,6] => ? = 2 - 1
[1,3,5,2,4,6,7] => [1,5,4,2,3,6,7] => [3,1,5,4,2,6,7] => [2,1,5,4,3,6,7] => ? = 3 - 1
[1,3,5,2,4,7,6] => [1,5,4,2,3,7,6] => [4,2,6,5,3,7,1] => [2,1,5,4,3,7,6] => ? = 1 - 1
[1,3,5,2,6,4,7] => [1,6,4,2,3,5,7] => [3,1,5,2,6,4,7] => [2,1,4,3,6,5,7] => ? = 2 - 1
[1,3,5,2,6,7,4] => [1,7,4,2,3,5,6] => [3,1,5,2,4,7,6] => [3,1,5,2,4,7,6] => ? = 1 - 1
[1,3,5,2,7,4,6] => [1,7,6,4,2,3,5] => [4,1,6,2,7,5,3] => [2,1,4,3,7,6,5] => ? = 1 - 1
[1,3,5,2,7,6,4] => [1,6,7,4,2,3,5] => [3,1,6,2,7,4,5] => [3,1,5,2,7,4,6] => ? = 2 - 1
[1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 4 - 1
[1,3,5,4,2,7,6] => [1,4,5,2,3,7,6] => [3,2,6,4,5,7,1] => [2,1,5,3,4,7,6] => ? = 1 - 1
[1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => [2,1,3,5,6,4,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
[1,3,5,4,6,7,2] => [1,4,7,2,3,5,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 1 - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St000990
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> [1] => ? = 1
[1,2] => [1,2] => [1,0,1,0]
=> [2,1] => 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,2] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 7
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 2
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 3
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 3
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 4
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => ? = 2
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 2
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 2
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 2
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => ? = 3
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 2
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 2
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 3
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 3
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St000654
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1
[1,2] => [1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[2,3,1] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,2,3] => [1,3,4,2] => [3,1,2,4] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 1
[1,4,3,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => [2,1,3,4] => 1
[2,4,1,3] => [4,3,1,2] => [3,4,2,1] => [3,2,1,4] => 1
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 2
[3,1,4,2] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => [1,3,4,2] => 3
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,2,1] => [4,1,3,2] => [2,4,3,1] => [4,2,1,3] => 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,3,2] => [3,4,2,1] => [4,3,1,2] => [1,4,3,2] => 2
[4,2,1,3] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 3
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[4,3,2,1] => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,2,4,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => [3,1,2,4,5] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => [4,3,1,2,5] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,4,3,5,2] => [4,1,3,2,5] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,4,2,5,3] => [4,1,5,2,3] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [6,1,5,2,3,4,7] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [7,1,5,2,3,4,6] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [4,1,2,6,3,5,7] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => ? = 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,2,5,6,7,4,3] => [5,4,1,2,3,6,7] => ? = 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,2,5,6,7,3,4] => [5,6,1,2,3,4,7] => ? = 2
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,2,5,6,4,3,7] => [1,5,4,2,3,6,7] => ? = 2
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,2,5,6,4,7,3] => [5,1,4,2,3,6,7] => ? = 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => ? = 3
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => ? = 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => ? = 2
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,2,4,5,7,6,3] => [7,4,1,2,3,5,6] => ? = 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,2,6,7,5,4,3] => [6,5,4,1,2,3,7] => ? = 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,2,6,7,5,3,4] => [6,7,5,1,2,3,4] => ? = 2
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,2,6,7,3,5,4] => [6,5,7,1,2,3,4] => ? = 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => ? = 3
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,2,6,7,4,5,3] => [6,4,7,1,2,3,5] => ? = 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,2,6,7,4,3,5] => [6,7,4,1,2,3,5] => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [5,1,7,4,2,3,6] => ? = 1
[1,2,5,3,6,4,7] => [1,2,6,4,3,5,7] => [1,2,5,4,6,3,7] => [1,5,2,4,3,6,7] => ? = 2
[1,2,5,3,6,7,4] => [1,2,7,4,3,5,6] => [1,2,5,4,6,7,3] => [5,1,2,4,3,6,7] => ? = 1
[1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => [1,2,6,5,7,4,3] => [6,5,1,4,2,3,7] => ? = 1
[1,2,5,3,7,6,4] => [1,2,6,7,4,3,5] => [1,2,6,5,7,3,4] => [6,7,1,5,2,3,4] => ? = 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => ? = 1
[1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => ? = 1
[1,2,5,4,7,3,6] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [6,5,1,7,2,3,4] => ? = 1
[1,2,5,4,7,6,3] => [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,6,2,7,3,4,5] => ? = 2
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St001184
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 3
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 3
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 3
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
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!