searching the database
Your data matches 14 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: St000142
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 0
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 0
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 0
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 0
[3,2,2]
=> 2
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 0
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of even parts of a partition.
Matching statistic: St000148
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 65%●distinct values known / distinct values provided: 44%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 65%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> []
=> []
=> 0
[2]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1]
=> [2]
=> []
=> []
=> 0
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [3]
=> []
=> []
=> 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[1,1,1,1]
=> [4]
=> []
=> []
=> 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[]
=> []
=> ?
=> ?
=> ? = 0
[7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 3
[6,6,5,4,3,2,1]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 4
[6,5,5,4,3,2,1]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 3
[5,5,5,4,3,2,1]
=> [7,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? = 2
[6,5,4,4,3,2,1]
=> [7,6,5,4,2,1]
=> [6,5,4,2,1]
=> [5,4,3,3,2,1]
=> ? = 4
[6,4,4,4,3,2,1]
=> [7,6,5,4,1,1]
=> [6,5,4,1,1]
=> [5,3,3,3,2,1]
=> ? = 5
[6,5,4,3,3,2,1]
=> [7,6,5,3,2,1]
=> [6,5,3,2,1]
=> [5,4,3,2,2,1]
=> ? = 3
[7,6,5,4,3,2]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 3
[7,6,2,2,2,2]
=> [6,6,2,2,2,2,1]
=> ?
=> ?
=> ? = 5
[4,4,4,4,3]
=> [5,5,5,4]
=> ?
=> ?
=> ? = 4
[4,4,4,3,3]
=> ?
=> ?
=> ?
=> ? = 3
[4,4,4,4,2]
=> [5,5,4,4]
=> ?
=> ?
=> ? = 5
[6,6,5,4]
=> [4,4,4,4,3,2]
=> ?
=> ?
=> ? = 3
[6,4,4,4]
=> [4,4,4,4,1,1]
=> ?
=> ?
=> ? = 4
[2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ?
=> ? = 10
[3,3,3,3,3,3,2]
=> ?
=> ?
=> ?
=> ? = 1
[4,4,3,3,3,3]
=> ?
=> ?
=> ?
=> ? = 2
[3,3,3,3,3,3,3,3,3]
=> ?
=> ?
=> ?
=> ? = 0
[4,4,4,4,3,3]
=> ?
=> ?
=> ?
=> ? = 4
[4,4,4,3,3,3]
=> [6,6,6,3]
=> ?
=> ?
=> ? = 3
[4,4,4,4,4,4]
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 6
[4,4,4,4,4,4,4,4]
=> ?
=> ?
=> ?
=> ? = 8
[18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ?
=> ? = 1
[19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ?
=> ? = 0
[20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ?
=> ? = 1
[21]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ?
=> ? = 0
[6,6,6,6]
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 4
[3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [16,16,1]
=> ?
=> ?
=> ? = 15
[7,6,6,6]
=> [4,4,4,4,4,4,1]
=> ?
=> ?
=> ? = 3
[7,2,2,2,2,2,2]
=> [7,7,1,1,1,1,1]
=> ?
=> ?
=> ? = 6
[4,2,2,2,2,2,2,2]
=> [8,8,1,1]
=> ?
=> ?
=> ? = 8
[10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 2
[6,6,6,6,6]
=> [5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5,5]
=> ? = 5
[11,7,3]
=> [3,3,3,2,2,2,2,1,1,1,1]
=> ?
=> ?
=> ? = 0
[3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [32,32,1]
=> ?
=> ?
=> ? = 31
[7,6,6,6,6,6,6,6]
=> [8,8,8,8,8,8,1]
=> ?
=> ?
=> ? = 7
[7,6,6,6,2,2,2,2,2,2,2,2]
=> [12,12,4,4,4,4,1]
=> ?
=> ?
=> ? = 11
[7,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> [14,14,2,2,2,2,1]
=> ?
=> ?
=> ? = 13
[7,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [15,15,1,1,1,1,1]
=> ?
=> ?
=> ? = 14
[4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [16,16,1,1]
=> ?
=> ?
=> ? = 16
[5,2,2,2,2,2,2,2]
=> [8,8,1,1,1]
=> ?
=> ?
=> ? = 7
[6,6,6,4,3,3]
=> [6,6,6,4,3,3]
=> ?
=> ?
=> ? = 4
[5,4,2,2,2,2,2,2]
=> [8,8,2,2,1]
=> ?
=> ?
=> ? = 7
[7,6,6,6,6]
=> [5,5,5,5,5,5,1]
=> ?
=> ?
=> ? = 4
[6,6,4,3,3,2,2]
=> [7,7,5,3,2,2]
=> ?
=> ?
=> ? = 5
[8,6,2,2,2,2]
=> [6,6,2,2,2,2,1,1]
=> ?
=> ?
=> ? = 6
[15,2,2,2,2]
=> [5,5,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ?
=> ? = 4
[5,4,4,4,2,2,2,2]
=> [8,8,4,4,1]
=> ?
=> ?
=> ? = 7
[7,6,6,6,6,6]
=> [6,6,6,6,6,6,1]
=> ?
=> ?
=> ? = 5
Description
The number of odd parts of a partition.
Matching statistic: St000992
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 59%●distinct values known / distinct values provided: 37%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 59%●distinct values known / distinct values provided: 37%
Values
[1]
=> [1]
=> []
=> 0
[2]
=> [1,1]
=> [1]
=> 1
[1,1]
=> [2]
=> []
=> 0
[3]
=> [1,1,1]
=> [1,1]
=> 0
[2,1]
=> [2,1]
=> [1]
=> 1
[1,1,1]
=> [3]
=> []
=> 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2]
=> [2,2]
=> [2]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> 1
[1,1,1,1]
=> [4]
=> []
=> 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [5]
=> []
=> 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> []
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 2
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[9,6]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 1
[8,7]
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> ? = 1
[16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[15,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> ? = 2
[10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 2
[8,8]
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 2
[8,6,2]
=> [3,3,2,2,2,2,1,1]
=> [3,2,2,2,2,1,1]
=> ? = 3
[8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> ? = 1
[7,6,3]
=> [3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> ? = 1
[6,6,4]
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> ? = 3
[6,5,5]
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> ? = 1
[17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[9,8]
=> [2,2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,2,1]
=> ? = 1
[8,8,1]
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 2
[8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [3,3,2,2,2,1,1]
=> ? = 2
[7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> ? = 1
[5,4,4,4]
=> [4,4,4,4,1]
=> [4,4,4,1]
=> ? = 3
[]
=> []
=> ?
=> ? = 0
[6,6,5,4,3,2,1]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 4
[6,5,5,4,3,2,1]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> ? = 3
[5,5,5,4,3,2,1]
=> [7,6,5,4,3]
=> [6,5,4,3]
=> ? = 2
[6,5,4,4,3,2,1]
=> [7,6,5,4,2,1]
=> [6,5,4,2,1]
=> ? = 4
[6,4,4,4,3,2,1]
=> [7,6,5,4,1,1]
=> [6,5,4,1,1]
=> ? = 5
[5,4,4,4,3,2,1]
=> [7,6,5,4,1]
=> [6,5,4,1]
=> ? = 4
[4,4,4,4,3,2,1]
=> [7,6,5,4]
=> [6,5,4]
=> ? = 5
[6,5,4,3,3,2,1]
=> [7,6,5,3,2,1]
=> [6,5,3,2,1]
=> ? = 3
[6,5,3,3,3,2,1]
=> [7,6,5,2,2,1]
=> [6,5,2,2,1]
=> ? = 2
[6,4,3,3,3,2,1]
=> [7,6,5,2,1,1]
=> [6,5,2,1,1]
=> ? = 3
[6,3,3,3,3,2,1]
=> [7,6,5,1,1,1]
=> [6,5,1,1,1]
=> ? = 2
[6,5,4,3,2,2,1]
=> [7,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 4
[6,5,4,2,2,2,1]
=> [7,6,3,3,2,1]
=> [6,3,3,2,1]
=> ? = 5
[6,5,3,2,2,2,1]
=> [7,6,3,2,2,1]
=> [6,3,2,2,1]
=> ? = 4
[6,5,2,2,2,2,1]
=> [7,6,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 5
[7,6,2,2,2,2]
=> [6,6,2,2,2,2,1]
=> ?
=> ? = 5
[6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> ? = 6
[4,4,4,4,3]
=> [5,5,5,4]
=> ?
=> ? = 4
[4,4,4,3,3]
=> ?
=> ?
=> ? = 3
[4,4,4,4,2]
=> [5,5,4,4]
=> ?
=> ? = 5
[7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> [4,4,3,2,1,1]
=> ? = 1
[6,6,5,4]
=> [4,4,4,4,3,2]
=> ?
=> ? = 3
[5,5,4,4]
=> [4,4,4,4,2]
=> [4,4,4,2]
=> ? = 2
[6,4,4,4]
=> [4,4,4,4,1,1]
=> ?
=> ? = 4
[3,3,3,3,3,3,3,3]
=> [8,8,8]
=> [8,8]
=> ? = 0
[2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 10
[3,3,3,3,3,3,2]
=> ?
=> ?
=> ? = 1
[3,3,3,3,3,3,3]
=> [7,7,7]
=> [7,7]
=> ? = 0
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000288
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Values
[1]
=> 1 => 0 => 0
[2]
=> 0 => 1 => 1
[1,1]
=> 11 => 00 => 0
[3]
=> 1 => 0 => 0
[2,1]
=> 01 => 10 => 1
[1,1,1]
=> 111 => 000 => 0
[4]
=> 0 => 1 => 1
[3,1]
=> 11 => 00 => 0
[2,2]
=> 00 => 11 => 2
[2,1,1]
=> 011 => 100 => 1
[1,1,1,1]
=> 1111 => 0000 => 0
[5]
=> 1 => 0 => 0
[4,1]
=> 01 => 10 => 1
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 000 => 0
[2,2,1]
=> 001 => 110 => 2
[2,1,1,1]
=> 0111 => 1000 => 1
[1,1,1,1,1]
=> 11111 => 00000 => 0
[6]
=> 0 => 1 => 1
[5,1]
=> 11 => 00 => 0
[4,2]
=> 00 => 11 => 2
[4,1,1]
=> 011 => 100 => 1
[3,3]
=> 11 => 00 => 0
[3,2,1]
=> 101 => 010 => 1
[3,1,1,1]
=> 1111 => 0000 => 0
[2,2,2]
=> 000 => 111 => 3
[2,2,1,1]
=> 0011 => 1100 => 2
[2,1,1,1,1]
=> 01111 => 10000 => 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 0
[7]
=> 1 => 0 => 0
[6,1]
=> 01 => 10 => 1
[5,2]
=> 10 => 01 => 1
[5,1,1]
=> 111 => 000 => 0
[4,3]
=> 01 => 10 => 1
[4,2,1]
=> 001 => 110 => 2
[4,1,1,1]
=> 0111 => 1000 => 1
[3,3,1]
=> 111 => 000 => 0
[3,2,2]
=> 100 => 011 => 2
[3,2,1,1]
=> 1011 => 0100 => 1
[3,1,1,1,1]
=> 11111 => 00000 => 0
[2,2,2,1]
=> 0001 => 1110 => 3
[2,2,1,1,1]
=> 00111 => 11000 => 2
[2,1,1,1,1,1]
=> 011111 => 100000 => 1
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0
[8]
=> 0 => 1 => 1
[7,1]
=> 11 => 00 => 0
[6,2]
=> 00 => 11 => 2
[6,1,1]
=> 011 => 100 => 1
[5,3]
=> 11 => 00 => 0
[5,2,1]
=> 101 => 010 => 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 0000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => ? = 0
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ? = 0
[2,2,2,2,2,2,1,1,1,1]
=> 0000001111 => 1111110000 => ? = 6
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => 111100000000 => ? = 4
[]
=> ? => ? => ? = 0
[6,5,4,3,2,1]
=> ? => ? => ? = 3
[5,5,4,3,2,1]
=> ? => ? => ? = 2
[6,4,4,3,2,1]
=> ? => ? => ? = 4
[5,4,4,3,2,1]
=> ? => ? => ? = 3
[4,4,4,3,2,1]
=> ? => ? => ? = 4
[6,5,3,3,2,1]
=> ? => ? => ? = 2
[5,5,3,3,2,1]
=> ? => ? => ? = 1
[5,4,3,3,2,1]
=> ? => ? => ? = 2
[6,3,3,3,2,1]
=> ? => ? => ? = 2
[6,5,4,2,2,1]
=> ? => ? => ? = 4
[5,5,4,2,2,1]
=> ? => ? => ? = 3
[6,4,4,2,2,1]
=> ? => ? => ? = 5
[6,5,2,2,2,1]
=> ? => ? => ? = 4
[6,5,4,3,1,1]
=> ? => ? => ? = 2
[5,5,4,3,1,1]
=> ? => ? => ? = 1
[6,4,4,3,1,1]
=> ? => ? => ? = 3
[6,5,3,3,1,1]
=> ? => ? => ? = 1
[5,5,3,3,1,1]
=> ? => ? => ? = 0
[6,5,4,1,1,1]
=> ? => ? => ? = 2
[6,5,4,3,2]
=> ? => ? => ? = 3
[5,5,4,3,2]
=> ? => ? => ? = 2
[6,4,4,3,2]
=> ? => ? => ? = 4
[6,5,3,3,2]
=> ? => ? => ? = 2
[5,5,3,3,2]
=> ? => ? => ? = 1
[6,5,4,2,2]
=> ? => ? => ? = 4
[5,5,4,2,2]
=> ? => ? => ? = 3
[6,4,4,2,2]
=> ? => ? => ? = 5
[6,5,4,3,1]
=> ? => ? => ? = 2
[6,5,4,3]
=> ? => ? => ? = 2
[7,6,5,4,3,2,1]
=> ? => ? => ? = 3
[6,6,5,4,3,2,1]
=> ? => ? => ? = 4
[6,5,5,4,3,2,1]
=> ? => ? => ? = 3
[5,5,5,4,3,2,1]
=> ? => ? => ? = 2
[6,5,4,4,3,2,1]
=> ? => ? => ? = 4
[6,4,4,4,3,2,1]
=> ? => ? => ? = 5
[5,4,4,4,3,2,1]
=> ? => ? => ? = 4
[4,4,4,4,3,2,1]
=> ? => ? => ? = 5
[6,5,4,3,3,2,1]
=> ? => ? => ? = 3
[6,5,3,3,3,2,1]
=> ? => ? => ? = 2
[6,4,3,3,3,2,1]
=> ? => ? => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000093
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 53%●distinct values known / distinct values provided: 26%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 53%●distinct values known / distinct values provided: 26%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4 = 3 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 0 + 1
[8]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> 00011111 => [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2]
=> 0000000 => [8] => ([],8)
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> 00000000 => [9] => ([],9)
=> ? = 8 + 1
[2,2,2,2,2,2,1,1,1,1]
=> 0000001111 => [7,1,1,1,1] => ?
=> ? = 6 + 1
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => [5,1,1,1,1,1,1,1,1] => ?
=> ? = 4 + 1
[3,2,2,2,2,2,2,2]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[]
=> ? => ? => ?
=> ? = 0 + 1
[6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[5,5,4,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,4,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[5,4,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[4,4,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,3,3,2,1]
=> ? => ? => ?
=> ? = 1 + 1
[5,4,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,3,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,2,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[5,5,4,2,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,4,4,2,2,1]
=> ? => ? => ?
=> ? = 5 + 1
[6,5,2,2,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,4,3,1,1]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,4,3,1,1]
=> ? => ? => ?
=> ? = 1 + 1
[6,4,4,3,1,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,5,3,3,1,1]
=> ? => ? => ?
=> ? = 1 + 1
[5,5,3,3,1,1]
=> ? => ? => ?
=> ? = 0 + 1
[6,5,4,1,1,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,3,2]
=> ? => ? => ?
=> ? = 3 + 1
[5,5,4,3,2]
=> ? => ? => ?
=> ? = 2 + 1
[6,4,4,3,2]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,3,3,2]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,3,3,2]
=> ? => ? => ?
=> ? = 1 + 1
[6,5,4,2,2]
=> ? => ? => ?
=> ? = 4 + 1
[5,5,4,2,2]
=> ? => ? => ?
=> ? = 3 + 1
[6,4,4,2,2]
=> ? => ? => ?
=> ? = 5 + 1
[6,5,4,3,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,3]
=> ? => ? => ?
=> ? = 2 + 1
[7,6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[5,5,5,4,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 53%●distinct values known / distinct values provided: 26%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 53%●distinct values known / distinct values provided: 26%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4 = 3 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 0 + 1
[8]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> 00011111 => [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2]
=> 0000000 => [8] => ([],8)
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> 00000000 => [9] => ([],9)
=> ? = 8 + 1
[2,2,2,2,2,2,1,1,1,1]
=> 0000001111 => [7,1,1,1,1] => ?
=> ? = 6 + 1
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => [5,1,1,1,1,1,1,1,1] => ?
=> ? = 4 + 1
[3,2,2,2,2,2,2,2]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[]
=> ? => ? => ?
=> ? = 0 + 1
[6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[5,5,4,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,4,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[5,4,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[4,4,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,3,3,2,1]
=> ? => ? => ?
=> ? = 1 + 1
[5,4,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,3,3,3,2,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,2,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[5,5,4,2,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,4,4,2,2,1]
=> ? => ? => ?
=> ? = 5 + 1
[6,5,2,2,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,4,3,1,1]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,4,3,1,1]
=> ? => ? => ?
=> ? = 1 + 1
[6,4,4,3,1,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,5,3,3,1,1]
=> ? => ? => ?
=> ? = 1 + 1
[5,5,3,3,1,1]
=> ? => ? => ?
=> ? = 0 + 1
[6,5,4,1,1,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,3,2]
=> ? => ? => ?
=> ? = 3 + 1
[5,5,4,3,2]
=> ? => ? => ?
=> ? = 2 + 1
[6,4,4,3,2]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,3,3,2]
=> ? => ? => ?
=> ? = 2 + 1
[5,5,3,3,2]
=> ? => ? => ?
=> ? = 1 + 1
[6,5,4,2,2]
=> ? => ? => ?
=> ? = 4 + 1
[5,5,4,2,2]
=> ? => ? => ?
=> ? = 3 + 1
[6,4,4,2,2]
=> ? => ? => ?
=> ? = 5 + 1
[6,5,4,3,1]
=> ? => ? => ?
=> ? = 2 + 1
[6,5,4,3]
=> ? => ? => ?
=> ? = 2 + 1
[7,6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
[6,6,5,4,3,2,1]
=> ? => ? => ?
=> ? = 4 + 1
[6,5,5,4,3,2,1]
=> ? => ? => ?
=> ? = 3 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001337
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 48%●distinct values known / distinct values provided: 26%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 48%●distinct values known / distinct values provided: 26%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4 = 3 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[8]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,4]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[4,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> 0011111 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,1,1,1,1,1,1,1]
=> 01111111 => [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[5,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[4,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,3,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[2,2,2,1,1,1,1]
=> 0001111 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[6,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> 00011111 => [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[5,4,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,3,2,1,1,1]
=> 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,2,1,1,1,1,1]
=> 11011111 => [1,1,2,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,2,2,2,2,2,1]
=> 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 0 + 1
[5,5,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[3,3,3,3,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[2,2,2,2,2,2,2]
=> 0000000 => [8] => ([],8)
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 0 + 1
[6,5,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,4,3,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,3,3,2,1]
=> 111101 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> 00000000 => [9] => ([],9)
=> ? = 8 + 1
[2,2,2,2,2,2,1,1,1,1]
=> 0000001111 => [7,1,1,1,1] => ?
=> ? = 6 + 1
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => [5,1,1,1,1,1,1,1,1] => ?
=> ? = 4 + 1
[6,3,3,3,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,5,4,1,1,1]
=> 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,3,3,3,2,1]
=> 111101 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,2,2,2,2,2,2]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[]
=> ? => ? => ?
=> ? = 0 + 1
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 48%●distinct values known / distinct values provided: 26%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 48%●distinct values known / distinct values provided: 26%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4 = 3 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[8]
=> 0 => [2] => ([],2)
=> 2 = 1 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,4]
=> 00 => [3] => ([],3)
=> 3 = 2 + 1
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[4,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> 0011111 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,1,1,1,1,1,1,1]
=> 01111111 => [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[5,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[4,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,3,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[2,2,2,1,1,1,1]
=> 0001111 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[6,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> 00011111 => [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[5,4,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,3,2,1,1,1]
=> 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,2,1,1,1,1,1]
=> 11011111 => [1,1,2,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,2,2,2,2,2,1]
=> 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 0 + 1
[5,5,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[3,3,3,3,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[2,2,2,2,2,2,2]
=> 0000000 => [8] => ([],8)
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 0 + 1
[6,5,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,4,3,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,3,3,3,2,1]
=> 111101 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ?
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> 00000000 => [9] => ([],9)
=> ? = 8 + 1
[2,2,2,2,2,2,1,1,1,1]
=> 0000001111 => [7,1,1,1,1] => ?
=> ? = 6 + 1
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => [5,1,1,1,1,1,1,1,1] => ?
=> ? = 4 + 1
[6,3,3,3,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,5,4,1,1,1]
=> 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[5,3,3,3,2,1]
=> 111101 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,2,2,2,2,2,2]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[]
=> ? => ? => ?
=> ? = 0 + 1
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000149
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
St000149: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> 0
[2]
=> [2]
=> 1
[1,1]
=> [1,1]
=> 0
[3]
=> [2,1]
=> 0
[2,1]
=> [3]
=> 1
[1,1,1]
=> [1,1,1]
=> 0
[4]
=> [2,2]
=> 1
[3,1]
=> [2,1,1]
=> 0
[2,2]
=> [4]
=> 2
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [1,1,1,1]
=> 0
[5]
=> [2,2,1]
=> 0
[4,1]
=> [3,2]
=> 1
[3,2]
=> [4,1]
=> 1
[3,1,1]
=> [2,1,1,1]
=> 0
[2,2,1]
=> [5]
=> 2
[2,1,1,1]
=> [3,1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[6]
=> [2,2,2]
=> 1
[5,1]
=> [2,2,1,1]
=> 0
[4,2]
=> [4,2]
=> 2
[4,1,1]
=> [4,1,1]
=> 1
[3,3]
=> [3,2,1]
=> 0
[3,2,1]
=> [3,3]
=> 1
[3,1,1,1]
=> [2,1,1,1,1]
=> 0
[2,2,2]
=> [6]
=> 3
[2,2,1,1]
=> [5,1]
=> 2
[2,1,1,1,1]
=> [3,1,1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7]
=> [2,2,2,1]
=> 0
[6,1]
=> [3,2,2]
=> 1
[5,2]
=> [4,2,1]
=> 1
[5,1,1]
=> [2,2,1,1,1]
=> 0
[4,3]
=> [4,3]
=> 1
[4,2,1]
=> [5,2]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,1,1]
=> 0
[3,2,2]
=> [6,1]
=> 2
[3,2,1,1]
=> [3,3,1]
=> 1
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> 0
[2,2,2,1]
=> [7]
=> 3
[2,2,1,1,1]
=> [5,1,1]
=> 2
[2,1,1,1,1,1]
=> [3,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
[8]
=> [2,2,2,2]
=> 1
[7,1]
=> [2,2,2,1,1]
=> 0
[6,2]
=> [4,2,2]
=> 2
[6,1,1]
=> [4,2,1,1]
=> 1
[5,3]
=> [3,2,2,1]
=> 0
[5,2,1]
=> [3,3,2]
=> 1
[6,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 1
[10,1,1]
=> [4,2,2,2,1,1]
=> ? = 1
[5,5,2]
=> [6,3,2,1]
=> ? = 1
[10,3]
=> [4,3,2,2,2]
=> ? = 1
[7,5,1]
=> [4,3,2,2,1,1]
=> ? = 0
[5,3,2,1,1,1]
=> [4,3,3,1,1,1]
=> ? = 1
[3,3,2,1,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> ? = 1
[13,1]
=> [2,2,2,2,2,2,1,1]
=> ? = 0
[12,2]
=> [4,2,2,2,2,2]
=> ? = 2
[7,5,2]
=> [6,3,2,2,1]
=> ? = 1
[5,5,4]
=> [6,5,2,1]
=> ? = 1
[5,5,1,1,1,1]
=> [4,3,2,1,1,1,1,1]
=> ? = 0
[5,4,2,1,1,1]
=> [7,4,1,1,1]
=> ? = 2
[5,2,2,2,2,1]
=> [9,3,2]
=> ? = 4
[3,3,3,3,1,1]
=> [3,3,3,2,1,1,1]
=> ? = 0
[14,1]
=> [3,2,2,2,2,2,2]
=> ? = 1
[9,6]
=> [4,4,4,2,1]
=> ? = 1
[9,5,1]
=> [4,3,2,2,2,1,1]
=> ? = 0
[8,5,2]
=> [6,4,3,2]
=> ? = 2
[7,5,3]
=> [4,3,3,2,2,1]
=> ? = 0
[6,5,1,1,1,1]
=> [6,3,2,1,1,1,1]
=> ? = 1
[5,4,3,3]
=> [5,5,4,1]
=> ? = 1
[5,4,3,1,1,1]
=> [5,5,2,1,1,1]
=> ? = 1
[5,3,2,2,2,1]
=> [7,3,3,2]
=> ? = 3
[4,4,4,1,1,1]
=> [9,4,1,1]
=> ? = 3
[4,3,3,3,2]
=> [7,5,2,1]
=> ? = 2
[15,1]
=> [2,2,2,2,2,2,2,1,1]
=> ? = 0
[8,6,2]
=> [6,4,4,2]
=> ? = 3
[8,5,3]
=> [5,4,4,2,1]
=> ? = 1
[7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 0
[6,5,5]
=> [6,5,4,1]
=> ? = 1
[6,4,3,3]
=> [7,6,2,1]
=> ? = 2
[5,5,2,2,2]
=> [10,3,2,1]
=> ? = 3
[5,4,3,2,1,1]
=> [5,5,5,1]
=> ? = 2
[5,4,2,2,2,1]
=> [10,3,3]
=> ? = 4
[4,4,3,3,2]
=> [9,6,1]
=> ? = 3
[4,3,3,3,3]
=> [5,5,3,2,1]
=> ? = 1
[2,2,2,2,2,2,1,1,1,1]
=> [13,1,1,1]
=> ? = 6
[17]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 0
[9,8]
=> [4,4,4,4,1]
=> ? = 1
[8,6,3]
=> [6,5,4,2]
=> ? = 2
[7,5,3,2]
=> [6,3,3,2,2,1]
=> ? = 1
[6,5,3,3]
=> [6,5,3,2,1]
=> ? = 1
[6,5,2,2,2]
=> [10,4,3]
=> ? = 4
[6,4,4,3]
=> [8,7,2]
=> ? = 3
[6,4,4,1,1,1]
=> [9,4,2,1,1]
=> ? = 3
[6,3,3,3,2]
=> [7,5,2,2,1]
=> ? = 2
[6,3,3,3,1,1]
=> [5,3,3,2,2,1,1]
=> ? = 1
[5,5,4,1,1,1]
=> [5,4,3,3,1,1]
=> ? = 1
[5,5,2,2,2,1]
=> [8,3,3,3]
=> ? = 3
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%
St000150: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 0
[3]
=> [3]
=> 0
[2,1]
=> [1,1,1]
=> 1
[1,1,1]
=> [2,1]
=> 0
[4]
=> [2,2]
=> 1
[3,1]
=> [3,1]
=> 0
[2,2]
=> [1,1,1,1]
=> 2
[2,1,1]
=> [2,1,1]
=> 1
[1,1,1,1]
=> [4]
=> 0
[5]
=> [5]
=> 0
[4,1]
=> [2,2,1]
=> 1
[3,2]
=> [3,1,1]
=> 1
[3,1,1]
=> [3,2]
=> 0
[2,2,1]
=> [1,1,1,1,1]
=> 2
[2,1,1,1]
=> [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [4,1]
=> 0
[6]
=> [3,3]
=> 1
[5,1]
=> [5,1]
=> 0
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [2,2,2]
=> 1
[3,3]
=> [6]
=> 0
[3,2,1]
=> [3,1,1,1]
=> 1
[3,1,1,1]
=> [3,2,1]
=> 0
[2,2,2]
=> [1,1,1,1,1,1]
=> 3
[2,2,1,1]
=> [2,1,1,1,1]
=> 2
[2,1,1,1,1]
=> [4,1,1]
=> 1
[1,1,1,1,1,1]
=> [4,2]
=> 0
[7]
=> [7]
=> 0
[6,1]
=> [3,3,1]
=> 1
[5,2]
=> [5,1,1]
=> 1
[5,1,1]
=> [5,2]
=> 0
[4,3]
=> [3,2,2]
=> 1
[4,2,1]
=> [2,2,1,1,1]
=> 2
[4,1,1,1]
=> [2,2,2,1]
=> 1
[3,3,1]
=> [6,1]
=> 0
[3,2,2]
=> [3,1,1,1,1]
=> 2
[3,2,1,1]
=> [3,2,1,1]
=> 1
[3,1,1,1,1]
=> [4,3]
=> 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 3
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> 2
[2,1,1,1,1,1]
=> [4,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> 0
[8]
=> [4,4]
=> 1
[7,1]
=> [7,1]
=> 0
[6,2]
=> [3,3,1,1]
=> 2
[6,1,1]
=> [3,3,2]
=> 1
[5,3]
=> [5,3]
=> 0
[5,2,1]
=> [5,1,1,1]
=> 1
[8,2,2]
=> [4,4,1,1,1,1]
=> ? = 3
[6,2,2,1,1]
=> [3,3,2,1,1,1,1]
=> ? = 3
[9,2,2]
=> [9,1,1,1,1]
=> ? = 2
[7,4,2]
=> [7,2,2,1,1]
=> ? = 2
[5,4,4]
=> [5,2,2,2,2]
=> ? = 2
[3,3,2,1,1,1,1,1]
=> [6,4,1,1,1]
=> ? = 1
[8,4,2]
=> [4,4,2,2,1,1]
=> ? = 3
[7,5,2]
=> [7,5,1,1]
=> ? = 1
[7,4,3]
=> [7,3,2,2]
=> ? = 1
[6,4,2,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 4
[5,5,1,1,1,1]
=> [10,4]
=> ? = 0
[5,4,2,1,1,1]
=> [5,2,2,2,1,1,1]
=> ? = 2
[5,2,2,2,2,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 4
[4,4,4,2]
=> [2,2,2,2,2,2,1,1]
=> ? = 4
[4,3,2,2,2,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> ? = 4
[3,3,3,2,2,1]
=> [6,3,1,1,1,1,1]
=> ? = 2
[14,1]
=> [7,7,1]
=> ? = 1
[11,2,2]
=> [11,1,1,1,1]
=> ? = 2
[9,6]
=> [9,3,3]
=> ? = 1
[8,7]
=> [7,4,4]
=> ? = 1
[8,5,2]
=> [5,4,4,1,1]
=> ? = 2
[6,5,4]
=> [5,3,3,2,2]
=> ? = 2
[6,2,2,2,2,1]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 5
[5,5,5]
=> [10,5]
=> ? = 0
[5,4,3,3]
=> [6,5,2,2]
=> ? = 1
[4,4,4,3]
=> [3,2,2,2,2,2,2]
=> ? = 3
[4,3,3,3,2]
=> [6,3,2,2,1,1]
=> ? = 2
[3,3,3,3,2,1]
=> [12,1,1,1]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4,2,1]
=> ? = 0
[8,6,2]
=> [4,4,3,3,1,1]
=> ? = 3
[7,6,3]
=> [7,3,3,3]
=> ? = 1
[6,5,5]
=> [10,3,3]
=> ? = 1
[6,4,3,3]
=> [6,3,3,2,2]
=> ? = 2
[5,5,2,2,2]
=> [10,1,1,1,1,1,1]
=> ? = 3
[5,4,4,3]
=> [5,3,2,2,2,2]
=> ? = 2
[5,4,3,2,1,1]
=> [5,3,2,2,2,1,1]
=> ? = 2
[5,4,2,2,2,1]
=> [5,2,2,1,1,1,1,1,1,1]
=> ? = 4
[4,4,3,3,2]
=> [6,2,2,2,2,1,1]
=> ? = 3
[4,3,3,3,3]
=> [12,2,2]
=> ? = 1
[4,3,3,3,2,1]
=> [6,3,2,2,1,1,1]
=> ? = 2
[2,2,2,2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 6
[2,2,2,2,1,1,1,1,1,1,1,1]
=> [8,1,1,1,1,1,1,1,1]
=> ? = 4
[9,8]
=> [9,4,4]
=> ? = 1
[8,8,1]
=> [4,4,4,4,1]
=> ? = 2
[7,5,3,2]
=> [7,5,3,1,1]
=> ? = 1
[6,5,2,2,2]
=> [5,3,3,1,1,1,1,1,1]
=> ? = 4
[6,4,4,3]
=> [3,3,3,2,2,2,2]
=> ? = 3
[6,4,4,1,1,1]
=> [3,3,2,2,2,2,2,1]
=> ? = 3
[5,5,4,3]
=> [10,3,2,2]
=> ? = 1
[5,5,4,1,1,1]
=> [10,2,2,2,1]
=> ? = 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000389The number of runs of ones of odd length in a binary word. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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!