Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000708
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => [1,2] => [2,1]
 => 2
[2]
 => 100 => [1,3] => [3,1]
 => 3
[1,1]
 => 110 => [1,1,2] => [2,1,1]
 => 2
[3]
 => 1000 => [1,4] => [4,1]
 => 4
[2,1]
 => 1010 => [1,2,2] => [2,2,1]
 => 4
[1,1,1]
 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 2
[4]
 => 10000 => [1,5] => [5,1]
 => 5
[3,1]
 => 10010 => [1,3,2] => [3,2,1]
 => 6
[2,2]
 => 1100 => [1,1,3] => [3,1,1]
 => 3
[2,1,1]
 => 10110 => [1,2,1,2] => [2,2,1,1]
 => 4
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
 => 2
[5]
 => 100000 => [1,6] => [6,1]
 => 6
[4,1]
 => 100010 => [1,4,2] => [4,2,1]
 => 8
[3,2]
 => 10100 => [1,2,3] => [3,2,1]
 => 6
[3,1,1]
 => 100110 => [1,3,1,2] => [3,2,1,1]
 => 6
[2,2,1]
 => 11010 => [1,1,2,2] => [2,2,1,1]
 => 4
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [2,2,1,1,1]
 => 4
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
 => 2
[6]
 => 1000000 => [1,7] => [7,1]
 => 7
[5,1]
 => 1000010 => [1,5,2] => [5,2,1]
 => 10
[4,2]
 => 100100 => [1,3,3] => [3,3,1]
 => 9
[4,1,1]
 => 1000110 => [1,4,1,2] => [4,2,1,1]
 => 8
[3,3]
 => 11000 => [1,1,4] => [4,1,1]
 => 4
[3,2,1]
 => 101010 => [1,2,2,2] => [2,2,2,1]
 => 8
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
 => 6
[2,2,2]
 => 11100 => [1,1,1,3] => [3,1,1,1]
 => 3
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [2,2,1,1,1]
 => 4
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
 => 2
[7]
 => 10000000 => [1,8] => [8,1]
 => 8
[6,1]
 => 10000010 => [1,6,2] => [6,2,1]
 => 12
[5,2]
 => 1000100 => [1,4,3] => [4,3,1]
 => 12
[5,1,1]
 => 10000110 => [1,5,1,2] => [5,2,1,1]
 => 10
[4,3]
 => 101000 => [1,2,4] => [4,2,1]
 => 8
[4,2,1]
 => 1001010 => [1,3,2,2] => [3,2,2,1]
 => 12
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
 => 8
[3,3,1]
 => 110010 => [1,1,3,2] => [3,2,1,1]
 => 6
[3,2,2]
 => 101100 => [1,2,1,3] => [3,2,1,1]
 => 6
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
 => 8
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
 => 6
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [2,2,1,1,1]
 => 4
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
 => 4
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
 => 4
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
 => 2
[8]
 => 100000000 => [1,9] => [9,1]
 => 9
[7,1]
 => 100000010 => [1,7,2] => [7,2,1]
 => 14
[6,2]
 => 10000100 => [1,5,3] => [5,3,1]
 => 15
[6,1,1]
 => 100000110 => [1,6,1,2] => [6,2,1,1]
 => 12
[5,3]
 => 1001000 => [1,3,4] => [4,3,1]
 => 12
[5,2,1]
 => 10001010 => [1,4,2,2] => [4,2,2,1]
 => 16
Description
The product of the parts of an integer partition.
Matching statistic: St001814
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001814: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 83%●distinct values known / distinct values provided: 81%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => 3
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 4
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 4
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 6
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 4
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 8
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 6
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 6
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 4
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 4
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 10
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 9
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 8
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 4
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 8
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 6
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => 2
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7]
 => 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,1,1,1,1]
 => 12
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 12
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1,1]
 => 10
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 8
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 12
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 8
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 6
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 6
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 8
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => 6
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => 4
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1,1,1]
 => 4
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => 2
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8]
 => 9
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,1,1,1,1,1]
 => ? = 14
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2]
 => 15
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,1,1]
 => 12
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 12
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 16
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1]
 => 10
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,1,1,1,1,1,1]
 => ? = 16
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,2,2,2,2]
 => ? = 18
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,1,1,1,1,1,1,1]
 => ? = 18
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3]
 => ? = 20
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,2,2,2,2,1]
 => ? = 24
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1,1]
 => ? = 20
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2,1]
 => ? = 12
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,2,2,2,2,1]
 => ? = 8
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,2,2,2,2,2,2]
 => ? = 24
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4]
 => ? = 20
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [8,4,4,4,4]
 => ? = 25
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [7,5,5]
 => ? = 18
[4,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2]
 => ? = 9
[9,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [9,4,4,4,4,4]
 => ? = 30
[8,5]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => [8,5,5,5]
 => ? = 24
[8,4,1]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? = 40
[3,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,2,2,1,1,1,1]
 => ? = 8
[8,5,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 40
[8,4,2]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? = 45
[7,4,2,1]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,2,2,1]
 => ? = 48
[5,4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [5,4,4,1]
 => ? = 16
[4,3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,1]
 => ? = 12
[4,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,2]
 => ? = 12
[4,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,3,3,2,2,1]
 => ? = 16
[8,5,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 48
[8,4,3]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [8,4,4,4,4,3]
 => ? = 40
[7,5,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,5,2,2,2,1]
 => ? = 48
[7,4,3,1]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,1,1]
 => ? = 48
[5,4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2]
 => ? = 18
[4,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,2]
 => ? = 12
[4,3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,3,3,2,1,1,1]
 => ? = 16
[4,3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,3,2,2,2,1]
 => ? = 16
[3,3,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ?
 => ? = 8
[3,3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ?
 => ? = 8
[8,6,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ?
 => ? = 45
[8,5,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? = 48
[7,5,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,1,1]
 => ? = 54
[7,4,3,2]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2]
 => ? = 48
[5,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,1]
 => ? = 24
[5,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1]
 => ? = 24
[5,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => ? = 24
[4,3,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,3,2,2,1,1]
 => ? = 16
[3,3,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ?
 => ?
 => ? = 8
[3,3,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ?
 => ? = 8
[8,6,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
 => ?
 => ?
 => ? = 48
[7,6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,1,1]
 => ? = 48
[7,5,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,1,1,1]
 => ? = 48
[7,5,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,2]
 => ? = 54
[5,4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,1]
 => ? = 24
Description
The number of partitions interlacing the given partition.
Matching statistic: St001232
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 23%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 6 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 4 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 8 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 6 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 10 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 9 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 8 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? = 6 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? = 4 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 8 - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 12 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 12 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 10 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 8 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 12 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 8 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 6 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 8 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 6 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 4 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 14 - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 15 - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 12 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 12 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 16 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 10 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 12 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 9 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 12 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 8 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 6 - 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 6 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 8 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 8 - 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 6 - 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 4 - 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.