Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001274
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => 0
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St001556
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1 = 0 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 1 = 0 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1 = 0 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1 = 0 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 1 = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ? = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 1 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ? = 0 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1 = 0 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1 = 0 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ? = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 2 = 1 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ? = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 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]
 => [2,3,4,5,6,7,8,1] => ? = 0 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 0 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ? = 0 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ? = 0 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => ? = 2 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2 = 1 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 1 = 0 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 2 = 1 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => ? = 2 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => ? = 1 + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => ? = 0 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => ? = 0 + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => ? = 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]
 => [3,2,4,5,6,7,8,1] => ? = 0 + 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]
 => [2,3,4,5,6,7,8,9,1] => ? = 0 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ? = 0 + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => ? = 0 + 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => ? = 0 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1,3,5] => ? = 0 + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 1 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => ? = 0 + 1
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => ? = 2 + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [5,6,2,1,3,4] => ? = 0 + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 2 = 1 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 1 = 0 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2 = 1 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => ? = 2 + 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => ? = 1 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => ? = 0 + 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2 = 1 + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => ? = 2 + 1
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => ? = 0 + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => ? = 0 + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => ? = 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]
 => [4,2,3,5,6,7,8,1] => ? = 0 + 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => ? = 2 + 1
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => ? = 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]
 => [3,4,2,5,6,7,8,1] => ? = 0 + 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]
 => [3,2,4,5,6,7,8,9,1] => ? = 0 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ? = 0 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ? = 0 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ? = 0 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => ? = 0 + 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [7,3,4,1,2,5,6] => ? = 1 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => ? = 0 + 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => ? = 2 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1,3,4] => ? = 0 + 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1,2,5] => ? = 1 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,1,5] => ? = 0 + 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,1,5] => ? = 1 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 2 = 1 + 1
Description
The number of inversions of the third entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001221
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001221: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 0
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 0
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => ? = 0
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 0
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => ? = 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 0
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 0
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 0
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 0
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 0
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 0
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 0
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0]
 => ? = 0
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => ? = 2
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[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]
 => [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 0
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 0
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 0
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 2
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 0
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => ? = 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 0
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
Description
The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.
Matching statistic: St001816
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 1
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 2
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => ? = 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => ? = 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => ? = 0
[3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 2
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
 => ? = 0
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 1
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2
[4,4,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
[4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => ? = 1
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => ? = 0
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 1
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 2
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 1
[3,3,3]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => ? = 1
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 2
[3,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 0
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 1
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => ? = 0
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[9,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[8,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[10,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[9,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[11,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[10,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[12,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[11,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[13,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[12,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[14,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[13,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[15,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[14,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St000782
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 2 + 1
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 2 + 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => ? = 1 + 1
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 2 + 1
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 1 + 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => ? = 2 + 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => ? = 1 + 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => ? = 0 + 1
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 2 + 1
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => ? = 1 + 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => ? = 0 + 1
[3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 1 + 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => ? = 2 + 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => ? = 1 + 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => ? = 0 + 1
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => ? = 0 + 1
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
 => ? = 0 + 1
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 2 + 1
[4,4,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => ? = 0 + 1
[4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => ? = 1 + 1
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => ? = 0 + 1
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 1 + 1
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => ? = 2 + 1
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => ? = 1 + 1
[3,3,3]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => ? = 0 + 1
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => ? = 1 + 1
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => ? = 2 + 1
[3,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => ? = 0 + 1
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => ? = 0 + 1
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => ? = 1 + 1
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => ? = 0 + 1
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[9,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[8,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[10,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[9,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[11,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[10,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[12,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[11,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[13,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[12,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[14,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[13,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[15,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[14,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001722
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 2 + 1
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 2 + 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 + 1
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 2 + 1
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 1 + 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 2 + 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 + 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 0 + 1
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 2 + 1
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 1 + 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 0 + 1
[3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 1 + 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 2 + 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 + 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 + 1
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 + 1
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1011111110000000 => ? = 0 + 1
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 2 + 1
[4,4,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0 + 1
[4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 1 + 1
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 0 + 1
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 1 + 1
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 2 + 1
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 + 1
[3,3,3]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 1 + 1
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 2 + 1
[3,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 + 1
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 + 1
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 1 + 1
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 0 + 1
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[9,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[8,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[10,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[9,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[11,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[10,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[12,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[11,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[13,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[12,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[14,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[13,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[15,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[14,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001583
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 + 3
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 + 3
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 + 3
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 + 3
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 1 + 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 2 + 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 + 3
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ? = 0 + 3
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 + 3
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 1 + 3
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 0 + 3
[3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 1 + 3
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 2 + 3
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 + 3
[2,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 0 + 3
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 0 + 3
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 1 + 3
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ? = 0 + 3
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,9,2] => ? = 0 + 3
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 + 3
[4,4,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 3
[4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 1 + 3
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 0 + 3
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 1 + 3
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 2 + 3
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 + 3
[3,3,3]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 0 + 3
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? = 1 + 3
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 2 + 3
[3,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 0 + 3
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 0 + 3
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 1 + 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]
 => [3,1,4,5,6,7,8,2] => ? = 0 + 3
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[9,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[8,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[10,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[9,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[11,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[10,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[12,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[11,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[13,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[12,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[14,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[13,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[15,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[14,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.