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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St001066
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000319
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,1,2,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,2,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,2,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,2,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> ? = 5
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,2,1,1,1,1] => [5,2,1,1,1,1]
=> ? = 4
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> ? = 5
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,1,2,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,2,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,2,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,2,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> ? = 5
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,2,1,1,1,1] => [5,2,1,1,1,1]
=> ? = 4
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> ? = 5
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000931
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1,1] => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,6,1,1] => [1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1,1] => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,5,1,1,1] => [1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1,1,1] => [1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,5,1,1,1] => [1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,6,1,1,1] => [1,1,1,6,1] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,4,2,1,1] => [1,1,2,4,1] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,2,4,1,1] => [1,1,4,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,1,1,3,1,1] => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,1,1,3,1,1] => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1,1,3,1,1] => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,1,3,1,1,1] => [1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [2,1,3,1,1,1] => [1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,1,3,1,1,1] => [1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,1,4,1,1] => [1,1,4,1,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,2,1,1,1,1,1] => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,2,1,1,1,1,1] => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,2,1,1,1,1,1] => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? => ? => ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,2,2,1,1,1] => [1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [2,4,1,1,1] => [1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,5,1,1] => [1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,5,1,1] => [1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,1,1,2,1,1] => [1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,1,1,2,1,1] => [1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,1,2,1,1,1] => [1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [3,1,3,1,1] => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 3 - 1
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001172
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,3,6,7,2,4,5,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,4,6,2,3,7,5,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,2,3,7,4,8] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,4,8,6] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,1,6,7,4,5,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,3,5,1,6,7,4,8] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,8,4,6] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,3,6,1,7,4,5,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,8,5] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,3,6,7,1,4,5,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,1,4,8,5] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,7,1,8,4,5,6] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,4,1,5,7,8,3,6] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,4,1,6,7,3,5,8] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,7,3,8] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,1,3,8,6] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,6,1,3,7,8,5] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [2,4,6,7,1,8,3,5] => ?
=> ?
=> ? = 1 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,8,4,6] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [2,5,6,1,3,4,7,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,8,4] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,5,6,7,1,3,4,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,1,3,8,4,6] => ?
=> ?
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ?
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,4,1,7,2,8,5,6] => ?
=> ?
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,4,1,7,8,2,5,6] => ?
=> ?
=> ? = 3 - 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,8,6] => ?
=> ?
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,4,6,1,7,2,5,8] => ?
=> ?
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,4,7,1,2,8,5,6] => ?
=> ?
=> ? = 3 - 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St000052
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,5,6,4,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,4,8,6] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,4,6] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,4,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,6,3,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,7,3,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,3,8,6] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,2,4,5,7,8,3,6] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,3,5] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,3,8,5,6] => [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,2,4,7,8,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,3,4,7,8] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,5,6,3,7,4,8] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,3,7,8,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,2,5,6,7,3,8,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,3,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,3,4,8,6] => [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,5,7,3,8,4,6] => [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,3,4,5,8] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,6,7,3,8,4,5] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,3,4,5] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,2,5,7,8,6] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,4,2,7,8,5,6] => [.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,5,2,7,8,6] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,6,2,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,7,2,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St001483
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
St001483: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000118
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[[.,.],.],.]],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,.],[.,.]],.]],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,[[.,.],.]]],.],.],.]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,.],[[.,.],.]]],.],.],.]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[.,[[.,.],.]],[[.,.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[.,.],[[[[.,.],.],.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[[.,.],[.,.]],.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [[[[.,.],[[[.,.],[.,.]],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [[[.,[[[[.,.],.],[.,.]],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [[[[.,.],[[.,[[.,.],.]],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [[[.,[[[.,.],[[.,.],.]],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[[.,.],[.,.]]],.]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[[.,.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[[[.,.],.],[[.,.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[[.,.],[[[.,.],.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[[[.,.],.],.],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[[.,.],[[.,[.,.]],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[[.,.],[.,.]],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,[[.,.],.]],.]]],.],.]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[[.,.],[.,[[.,.],.]]]],.],.]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,.],[[.,.],.]]]],.],.]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[[[.,.],.],.]]]],.],.]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[.,[[.,.],.]],.],[[.,.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[[[.,.],.],.]],[[.,.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[.,[[.,.],.]]],[[.,.],.]],.]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[[[[.,.],.],.],[[[.,.],.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[[.,[[.,.],.]],[[[.,.],.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,2,3,1] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,2,3,1] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,5,1] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,2,1] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,2,1,3,1] => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,1,1,2,1] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,2,3,1] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,2,3,1] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,3,2,1] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,5,1] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,2,1] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,2,1,3,1] => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,1,1,2,1] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000365The number of double ascents of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000932The number of occurrences of the pattern UDU in a Dyck path. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. 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 $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001330The hat guessing number of a graph. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000028The number of stack-sorts needed to sort a permutation. St001644The dimension of a graph. St000308The height of the tree associated to a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000328The maximum number of child nodes in a tree. St000444The length of the maximal rise of a Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000982The length of the longest constant subword. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000983The length of the longest alternating subword. St001095The number of non-isomorphic posets with precisely one further covering relation. St000381The largest part of an integer composition.
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