searching the database
Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001940
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001195
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? = 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 0
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 0
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 0
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St000508
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,10],[2,9,11,12,13,14,15,16]]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
Description
Eigenvalues of the random-to-random operator acting on a simple module.
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module [1].
Matching statistic: St001001
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 101111111100000000 => ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 10111101100000 => ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1011111101000000 => ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 101111111100000000 => ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001557
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001730
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 101111111100000000 => ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 10111101100000 => ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1011111101000000 => ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 101111111100000000 => ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 - 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 - 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 - 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 - 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 - 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 1 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 0 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> ? = 0 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> ? = 1 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> ? = 0 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> ? = 0 - 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,10],[2,9,11,12,13,14,15,16]]
=> ? = 0 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 - 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001804
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 0 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0 + 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 + 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 0 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 + 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> ? = 0 + 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 + 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0 + 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0 + 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0 + 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> ? = 1 + 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> ? = 0 + 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,10],[2,9,11,12,13,14,15,16]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> ? = 0 + 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[7,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[8,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[8,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[7,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[9,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[9,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[8,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[10,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[10,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[9,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[11,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[11,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!