searching the database
Your data matches 78 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: St000755
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 2
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 2
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 2
[3,3]
=> 1
[3,2,1]
=> 1
[3,1,1,1]
=> 1
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 2
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 2
[4,2,1]
=> 2
[4,1,1,1]
=> 2
[3,3,1]
=> 1
[3,2,2]
=> 3
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 2
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001732
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[6]
=> []
=> []
=> []
=> ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[7]
=> []
=> []
=> []
=> ? = 3
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[8]
=> []
=> []
=> []
=> ? = 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[9]
=> []
=> []
=> []
=> ? ∊ {3,3}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3}
[10]
=> []
=> []
=> []
=> ? ∊ {2,2,3}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3}
[11]
=> []
=> []
=> []
=> ? ∊ {1,1,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3,3,3}
[12]
=> []
=> []
=> []
=> ? ∊ {1,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2}
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St001257
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> ?
=> ? = 1
[2]
=> []
=> []
=> ?
=> ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> ?
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4]
=> []
=> []
=> ?
=> ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[5]
=> []
=> []
=> ?
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[6]
=> []
=> []
=> ?
=> ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[7]
=> []
=> []
=> ?
=> ? ∊ {1,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,3}
[8]
=> []
=> []
=> ?
=> ? ∊ {2,2,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2}
[9]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,3,3}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3}
[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]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,2,3,3}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3}
[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]
=> ?
=> ? ∊ {1,1,2,3,3}
[10]
=> []
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3}
[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]
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3}
[11]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [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]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3}
[12]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1,1,1]
=> [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]
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000745
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> ?
=> ? = 1
[2]
=> []
=> []
=> ?
=> ? = 1
[1,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[3]
=> []
=> []
=> ?
=> ? = 1
[2,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[4]
=> []
=> []
=> ?
=> ? = 2
[3,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[2,2]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[5]
=> []
=> []
=> ?
=> ? = 1
[4,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[3,2]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[6]
=> []
=> []
=> ?
=> ? = 2
[5,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[4,2]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 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
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 1
[7]
=> []
=> []
=> ?
=> ? = 3
[6,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[5,2]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 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
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 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]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> 1
[8]
=> []
=> []
=> ?
=> ? ∊ {1,2}
[7,1]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 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
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 2
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? ∊ {1,2}
[9]
=> []
=> []
=> ?
=> ? ∊ {1,1,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> ? ∊ {1,1,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? ∊ {1,1,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14],[3,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,3,3}
[10]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,2,2,2,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12],[2,4,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12],[4,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12,14],[2,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14],[3,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14,16],[3,5,7,9,11,13,15,17,18]]
=> ? ∊ {1,1,1,1,2,2,2,3}
[11]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12],[2,4,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,3,4,5,8,10,12],[2,6,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12,14],[2,4,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12],[4,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12,14],[2,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14],[3,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,4,8,10,12],[5,6,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12,14],[4,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12,14,16],[2,5,7,9,11,13,15,17,18]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14,16],[3,5,7,9,11,13,15,17,18]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14,16,18],[3,5,7,9,11,13,15,17,19,20]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[12]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,5,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,9,10,12],[2,4,6,8,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12],[2,4,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,5,6,7,10,12],[2,4,8,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12,14],[2,4,6,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,3,4,5,8,10,12],[2,6,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12,14],[2,4,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12],[4,5,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12,14],[2,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12,14],[3,5,7,9,11,13,15,16]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,5,8,10,12],[4,6,7,9,11,13,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000701
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> ? ∊ {1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [[.,[.,[.,.]]],[[.,.],.]]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[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]
=> [1,9,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? ∊ {1,2,2}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ? ∊ {1,2,2,2,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> ? ∊ {1,2,2,2,3,3}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,2,2,2,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? ∊ {1,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? ∊ {1,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ? ∊ {1,2,2,2,3,3}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],[.,.]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [.,[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,1,2,3,4,5,6,7,8,11] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,9,2,3,4,5,6,7,10] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,1,2,3,4,5,6,8,10] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,8,3,4,5,6,9] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,1,2,3,4,5,9] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,1,2,3,4,6,7,9] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,2,3,9,4,5,6,7] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,2,3,4,5,7,8] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,2,3,4,5,6,7,9] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,2,3,4,5,6,7,8,9] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,1,2,3,4,5,6,7,8,9,12] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,10,2,3,4,5,6,7,8,11] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,1,2,3,4,5,6,7,9,11] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,9,3,4,5,6,7,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,9,1,2,3,4,5,6,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,1,2,3,4,5,7,8,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,8,4,5,6,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [6,7,1,8,2,3,4,5,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,8,2,3,4,5,7,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,1,2,3,4,6,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [4,8,1,2,3,5,6,7,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The protection number of a binary tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St001568
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,2}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2}
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,2,2,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,2}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2}
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,1,1,1]
=> [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]
=> [4]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,3}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,3}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,3}
[5,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,3}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,3}
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,3}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,3}
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2}
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,1]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[7,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[6,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[5,3,1]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[5,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[4,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[3,3,3]
=> [3,3]
=> [3]
=> [3]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,1]
=> 2
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> 1
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,2,3}
[9,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,2,3}
[8,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,2,3}
[8,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,3}
[7,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,2,3}
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000678
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000542
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [1,2,3,4,5] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [6,1,2,3,4,5] => 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,4,2] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,4] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [1,5,2,3,4] => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [5,6,1,2,3,4] => 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [4,1,5,2,3] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [1,5,2,3,4] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [1,4,2,3,5] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => [1,6,2,3,4,5] => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => ? ∊ {1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => ? ∊ {1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [5,1,6,2,3,4] => 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [4,6,1,2,3,5] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [4,1,2,5,3] => 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,4,5,1,2] => 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [4,1,2,3,5] => 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,2,2}
[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]
=> [1,9,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {1,1,2,2}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8] => ? ∊ {1,1,1,2,2,2,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [7,8,1,2,3,4,5,6] => ? ∊ {1,1,1,2,2,2,3,3}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [6,1,7,2,3,4,5] => ? ∊ {1,1,1,2,2,2,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [5,7,1,2,3,4,6] => ? ∊ {1,1,1,2,2,2,3,3}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,2,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {1,1,1,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {1,1,1,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,2,2,2,3,3}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [10,1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [8,9,1,2,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [7,1,8,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [6,8,1,2,3,4,5,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [6,1,2,7,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [4,7,1,2,3,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,2,3,8,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,2,3,4,5] => [1,6,7,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [1,7,2,3,4,5,6,8] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,1,2,3,4,5,6,7,8,11] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,9,2,3,4,5,6,7,10] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,1,2,3,4,5,6,8,10] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,8,3,4,5,6,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,1,2,3,4,5,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,1,2,3,4,6,7,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,1,7,2,3,4,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,1,2,3,5,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,1,2,4,5,6,8] => [3,7,1,2,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,2,8,3,4,5] => [1,6,7,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,2,3,9,4,5,6,7] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,2,3,4,6] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,2,3,4,5,6] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,2,3,4,5,7,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,2,3,4,5,6,7,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,2,3,4,5,6,7,8,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000864
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [6,1,2,3,4,5] => 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,4,2] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 3 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [5,6,1,2,3,4] => 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [4,1,5,2,3] => 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [1,5,2,3,4] => 0 = 1 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => [1,6,2,3,4,5] => 0 = 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]
=> [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => ? ∊ {1,1,2,2} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => ? ∊ {1,1,2,2} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [5,1,6,2,3,4] => 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [4,6,1,2,3,5] => 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [4,1,2,5,3] => 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,4,5,1,2] => 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [4,1,2,3,5] => 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,2,2} - 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]
=> [1,9,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {1,1,2,2} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [7,8,1,2,3,4,5,6] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [6,1,7,2,3,4,5] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [5,7,1,2,3,4,6] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,2,2,2,3,3} - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [10,1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [8,9,1,2,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [7,1,8,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [6,8,1,2,3,4,5,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [6,1,2,7,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [4,7,1,2,3,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,2,3,8,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,2,3,4,5] => [1,6,7,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [1,7,2,3,4,5,6,8] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3} - 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,1,2,3,4,5,6,7,8,11] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,9,2,3,4,5,6,7,10] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,1,2,3,4,5,6,8,10] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,8,3,4,5,6,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,1,2,3,4,5,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,1,2,3,4,6,7,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,1,7,2,3,4,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,1,2,3,5,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,1,2,4,5,6,8] => [3,7,1,2,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,2,8,3,4,5] => [1,6,7,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,2,3,9,4,5,6,7] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,2,3,4,6] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,2,3,4,5,6] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,2,3,4,5,7,8] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,2,3,4,5,6,7,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,2,3,4,5,6,7,8,9] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St001135
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
The following 68 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000552The number of cut vertices of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000007The number of saliances of the permutation. St000352The Elizalde-Pak rank of a permutation. St000068The number of minimal elements in a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000668The least common multiple of the parts of the partition. St000990The first ascent of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000092The number of outer peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000700The protection number of an ordered tree. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000353The number of inner valleys of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001114The number of odd descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000862The number of parts of the shifted shape of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000741The Colin de Verdière graph invariant. St000891The number of distinct diagonal sums of a permutation matrix. St001115The number of even descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001728The number of invisible descents of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000754The Grundy value for the game of removing nestings in a perfect matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000090The variation of a composition. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000764The number of strong records in an integer composition. St000768The number of peaks in an integer composition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
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!