Processing math: 100%

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000378
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> []
=> 0
1 => [1,1] => [1,0,1,0]
=> [1]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> []
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 5
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 7
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 6
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 5
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 9
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 9
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 12
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 6
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 11
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 14
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 6
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 8
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells c in the diagram of an integer partition λ for which arm(c)leg(c){0,1}. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000012
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St000012: Dyck paths ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 85%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
10000 => [1,5] => [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
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 8
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 8
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
00100000 => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 7
01100000 => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 6
10000000 => [1,8] => [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
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 21
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 36
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
000000010 => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
000000011 => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
000001000 => [6,4] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
000010000 => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
000100000 => [4,6] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
001000000 => [3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
010000000 => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 36
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 45
1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
Description
The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' (a1,,an) such that a1=0,ak+1ak+1. 2. The generating function Dn(q)=DDnqarea(D) satisfy the recurrence Dn+1(q)=qkDk(q)Dnk(q). 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of q,t-Catalan numbers.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000006: Dyck paths ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 62%
Values
0 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 9
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 9
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 12
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 6
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 11
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 8
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 4
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 8
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 9
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 9
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 11
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 13
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 12
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 11
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 14
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 10
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 12
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 9
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 14
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 18
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 11
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 13
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 17
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 16
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 15
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 20
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 8
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 7
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 11
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 10
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 13
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 10
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 12
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 16
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 15
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 14
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 19
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
Description
The dinv of a Dyck path. Let a=(a1,,an) be the area sequence of a Dyck path D (see [[St000012]]). The dinv statistic of D is dinv(D)=#{i<j:aiaj{0,1}}. Equivalently, dinv(D) is also equal to the number of boxes in the partition above D whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the (area,dinv) pair of statistics, see [2]. Let a=(0,a2,,ar,0,ar+2,,an) be the area sequence of the Dyck path D with ai>0 for 2ir (so that the path touches the diagonal for the first time after r steps). Assume that D has v entries where ai=0. Let D be the path with the area sequence (0,ar+2,,an,a21,a31,,ar1), then the statistics are related by (area(D),dinv(D))=(area(D)+r1,dinv(D)+v1).
Matching statistic: St001295
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St001295: Dyck paths ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 62%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
10000 => [1,5] => [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
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 8
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 8
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 7
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 9
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 9
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 11
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 13
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 8
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 12
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 11
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 14
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 7
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 10
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 12
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 9
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 15
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 14
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 18
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 6
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 11
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 13
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 17
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 10
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 16
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 15
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 20
100000 => [1,6] => [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
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 4
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 6
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 8
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 7
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 11
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 10
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 13
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 5
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 10
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 16
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 9
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 15
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 14
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 19
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 7
Description
Gives the vector space dimension of the homomorphism space between J^2 and J^2.