Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000142
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1
1 => [1,1] => [1,1]
 => 0
00 => [3] => [3]
 => 0
01 => [2,1] => [2,1]
 => 1
10 => [1,2] => [2,1]
 => 1
11 => [1,1,1] => [1,1,1]
 => 0
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 0
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 1
100 => [1,3] => [3,1]
 => 0
101 => [1,2,1] => [2,1,1]
 => 1
110 => [1,1,2] => [2,1,1]
 => 1
111 => [1,1,1,1] => [1,1,1,1]
 => 0
0000 => [5] => [5]
 => 0
0001 => [4,1] => [4,1]
 => 1
0010 => [3,2] => [3,2]
 => 1
0011 => [3,1,1] => [3,1,1]
 => 0
0100 => [2,3] => [3,2]
 => 1
0101 => [2,2,1] => [2,2,1]
 => 2
0110 => [2,1,2] => [2,2,1]
 => 2
0111 => [2,1,1,1] => [2,1,1,1]
 => 1
1000 => [1,4] => [4,1]
 => 1
1001 => [1,3,1] => [3,1,1]
 => 0
1010 => [1,2,2] => [2,2,1]
 => 2
1011 => [1,2,1,1] => [2,1,1,1]
 => 1
1100 => [1,1,3] => [3,1,1]
 => 0
1101 => [1,1,2,1] => [2,1,1,1]
 => 1
1110 => [1,1,1,2] => [2,1,1,1]
 => 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0
00000 => [6] => [6]
 => 1
00001 => [5,1] => [5,1]
 => 0
00010 => [4,2] => [4,2]
 => 2
00011 => [4,1,1] => [4,1,1]
 => 1
00100 => [3,3] => [3,3]
 => 0
00101 => [3,2,1] => [3,2,1]
 => 1
00110 => [3,1,2] => [3,2,1]
 => 1
00111 => [3,1,1,1] => [3,1,1,1]
 => 0
01000 => [2,4] => [4,2]
 => 2
01001 => [2,3,1] => [3,2,1]
 => 1
01010 => [2,2,2] => [2,2,2]
 => 3
01011 => [2,2,1,1] => [2,2,1,1]
 => 2
01100 => [2,1,3] => [3,2,1]
 => 1
01101 => [2,1,2,1] => [2,2,1,1]
 => 2
01110 => [2,1,1,2] => [2,2,1,1]
 => 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1
10000 => [1,5] => [5,1]
 => 0
10001 => [1,4,1] => [4,1,1]
 => 1
10010 => [1,3,2] => [3,2,1]
 => 1
10011 => [1,3,1,1] => [3,1,1,1]
 => 0
Description
The number of even parts of a partition.
Matching statistic: St000149
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => [2]
 => 1
1 => [1,1] => [1,1]
 => [1,1]
 => 0
00 => [3] => [3]
 => [2,1]
 => 0
01 => [2,1] => [2,1]
 => [3]
 => 1
10 => [1,2] => [2,1]
 => [3]
 => 1
11 => [1,1,1] => [1,1,1]
 => [1,1,1]
 => 0
000 => [4] => [4]
 => [2,2]
 => 1
001 => [3,1] => [3,1]
 => [2,1,1]
 => 0
010 => [2,2] => [2,2]
 => [4]
 => 2
011 => [2,1,1] => [2,1,1]
 => [3,1]
 => 1
100 => [1,3] => [3,1]
 => [2,1,1]
 => 0
101 => [1,2,1] => [2,1,1]
 => [3,1]
 => 1
110 => [1,1,2] => [2,1,1]
 => [3,1]
 => 1
111 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1,1]
 => 0
0000 => [5] => [5]
 => [2,2,1]
 => 0
0001 => [4,1] => [4,1]
 => [3,2]
 => 1
0010 => [3,2] => [3,2]
 => [4,1]
 => 1
0011 => [3,1,1] => [3,1,1]
 => [2,1,1,1]
 => 0
0100 => [2,3] => [3,2]
 => [4,1]
 => 1
0101 => [2,2,1] => [2,2,1]
 => [5]
 => 2
0110 => [2,1,2] => [2,2,1]
 => [5]
 => 2
0111 => [2,1,1,1] => [2,1,1,1]
 => [3,1,1]
 => 1
1000 => [1,4] => [4,1]
 => [3,2]
 => 1
1001 => [1,3,1] => [3,1,1]
 => [2,1,1,1]
 => 0
1010 => [1,2,2] => [2,2,1]
 => [5]
 => 2
1011 => [1,2,1,1] => [2,1,1,1]
 => [3,1,1]
 => 1
1100 => [1,1,3] => [3,1,1]
 => [2,1,1,1]
 => 0
1101 => [1,1,2,1] => [2,1,1,1]
 => [3,1,1]
 => 1
1110 => [1,1,1,2] => [2,1,1,1]
 => [3,1,1]
 => 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
00000 => [6] => [6]
 => [2,2,2]
 => 1
00001 => [5,1] => [5,1]
 => [2,2,1,1]
 => 0
00010 => [4,2] => [4,2]
 => [4,2]
 => 2
00011 => [4,1,1] => [4,1,1]
 => [4,1,1]
 => 1
00100 => [3,3] => [3,3]
 => [3,2,1]
 => 0
00101 => [3,2,1] => [3,2,1]
 => [3,3]
 => 1
00110 => [3,1,2] => [3,2,1]
 => [3,3]
 => 1
00111 => [3,1,1,1] => [3,1,1,1]
 => [2,1,1,1,1]
 => 0
01000 => [2,4] => [4,2]
 => [4,2]
 => 2
01001 => [2,3,1] => [3,2,1]
 => [3,3]
 => 1
01010 => [2,2,2] => [2,2,2]
 => [6]
 => 3
01011 => [2,2,1,1] => [2,2,1,1]
 => [5,1]
 => 2
01100 => [2,1,3] => [3,2,1]
 => [3,3]
 => 1
01101 => [2,1,2,1] => [2,2,1,1]
 => [5,1]
 => 2
01110 => [2,1,1,2] => [2,2,1,1]
 => [5,1]
 => 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => [3,1,1,1]
 => 1
10000 => [1,5] => [5,1]
 => [2,2,1,1]
 => 0
10001 => [1,4,1] => [4,1,1]
 => [4,1,1]
 => 1
10010 => [1,3,2] => [3,2,1]
 => [3,3]
 => 1
10011 => [1,3,1,1] => [3,1,1,1]
 => [2,1,1,1,1]
 => 0
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
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => [1,1]
 => 1
1 => [1,1] => [1,1]
 => [2]
 => 0
00 => [3] => [3]
 => [3]
 => 0
01 => [2,1] => [2,1]
 => [1,1,1]
 => 1
10 => [1,2] => [2,1]
 => [1,1,1]
 => 1
11 => [1,1,1] => [1,1,1]
 => [2,1]
 => 0
000 => [4] => [4]
 => [2,2]
 => 1
001 => [3,1] => [3,1]
 => [3,1]
 => 0
010 => [2,2] => [2,2]
 => [1,1,1,1]
 => 2
011 => [2,1,1] => [2,1,1]
 => [2,1,1]
 => 1
100 => [1,3] => [3,1]
 => [3,1]
 => 0
101 => [1,2,1] => [2,1,1]
 => [2,1,1]
 => 1
110 => [1,1,2] => [2,1,1]
 => [2,1,1]
 => 1
111 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 0
0000 => [5] => [5]
 => [5]
 => 0
0001 => [4,1] => [4,1]
 => [2,2,1]
 => 1
0010 => [3,2] => [3,2]
 => [3,1,1]
 => 1
0011 => [3,1,1] => [3,1,1]
 => [3,2]
 => 0
0100 => [2,3] => [3,2]
 => [3,1,1]
 => 1
0101 => [2,2,1] => [2,2,1]
 => [1,1,1,1,1]
 => 2
0110 => [2,1,2] => [2,2,1]
 => [1,1,1,1,1]
 => 2
0111 => [2,1,1,1] => [2,1,1,1]
 => [2,1,1,1]
 => 1
1000 => [1,4] => [4,1]
 => [2,2,1]
 => 1
1001 => [1,3,1] => [3,1,1]
 => [3,2]
 => 0
1010 => [1,2,2] => [2,2,1]
 => [1,1,1,1,1]
 => 2
1011 => [1,2,1,1] => [2,1,1,1]
 => [2,1,1,1]
 => 1
1100 => [1,1,3] => [3,1,1]
 => [3,2]
 => 0
1101 => [1,1,2,1] => [2,1,1,1]
 => [2,1,1,1]
 => 1
1110 => [1,1,1,2] => [2,1,1,1]
 => [2,1,1,1]
 => 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => [4,1]
 => 0
00000 => [6] => [6]
 => [3,3]
 => 1
00001 => [5,1] => [5,1]
 => [5,1]
 => 0
00010 => [4,2] => [4,2]
 => [2,2,1,1]
 => 2
00011 => [4,1,1] => [4,1,1]
 => [2,2,2]
 => 1
00100 => [3,3] => [3,3]
 => [6]
 => 0
00101 => [3,2,1] => [3,2,1]
 => [3,1,1,1]
 => 1
00110 => [3,1,2] => [3,2,1]
 => [3,1,1,1]
 => 1
00111 => [3,1,1,1] => [3,1,1,1]
 => [3,2,1]
 => 0
01000 => [2,4] => [4,2]
 => [2,2,1,1]
 => 2
01001 => [2,3,1] => [3,2,1]
 => [3,1,1,1]
 => 1
01010 => [2,2,2] => [2,2,2]
 => [1,1,1,1,1,1]
 => 3
01011 => [2,2,1,1] => [2,2,1,1]
 => [2,1,1,1,1]
 => 2
01100 => [2,1,3] => [3,2,1]
 => [3,1,1,1]
 => 1
01101 => [2,1,2,1] => [2,2,1,1]
 => [2,1,1,1,1]
 => 2
01110 => [2,1,1,2] => [2,2,1,1]
 => [2,1,1,1,1]
 => 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => [4,1,1]
 => 1
10000 => [1,5] => [5,1]
 => [5,1]
 => 0
10001 => [1,4,1] => [4,1,1]
 => [2,2,2]
 => 1
10010 => [1,3,2] => [3,2,1]
 => [3,1,1,1]
 => 1
10011 => [1,3,1,1] => [3,1,1,1]
 => [3,2,1]
 => 0
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000237
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,1] => 1
1 => [1,1] => [1,0,1,0]
 => [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 0
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000389
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 1
1 => [1,1] => 11 => 00 => 0
00 => [3] => 100 => 011 => 0
01 => [2,1] => 101 => 010 => 1
10 => [1,2] => 110 => 001 => 1
11 => [1,1,1] => 111 => 000 => 0
000 => [4] => 1000 => 0111 => 1
001 => [3,1] => 1001 => 0110 => 0
010 => [2,2] => 1010 => 0101 => 2
011 => [2,1,1] => 1011 => 0100 => 1
100 => [1,3] => 1100 => 0011 => 0
101 => [1,2,1] => 1101 => 0010 => 1
110 => [1,1,2] => 1110 => 0001 => 1
111 => [1,1,1,1] => 1111 => 0000 => 0
0000 => [5] => 10000 => 01111 => 0
0001 => [4,1] => 10001 => 01110 => 1
0010 => [3,2] => 10010 => 01101 => 1
0011 => [3,1,1] => 10011 => 01100 => 0
0100 => [2,3] => 10100 => 01011 => 1
0101 => [2,2,1] => 10101 => 01010 => 2
0110 => [2,1,2] => 10110 => 01001 => 2
0111 => [2,1,1,1] => 10111 => 01000 => 1
1000 => [1,4] => 11000 => 00111 => 1
1001 => [1,3,1] => 11001 => 00110 => 0
1010 => [1,2,2] => 11010 => 00101 => 2
1011 => [1,2,1,1] => 11011 => 00100 => 1
1100 => [1,1,3] => 11100 => 00011 => 0
1101 => [1,1,2,1] => 11101 => 00010 => 1
1110 => [1,1,1,2] => 11110 => 00001 => 1
1111 => [1,1,1,1,1] => 11111 => 00000 => 0
00000 => [6] => 100000 => 011111 => 1
00001 => [5,1] => 100001 => 011110 => 0
00010 => [4,2] => 100010 => 011101 => 2
00011 => [4,1,1] => 100011 => 011100 => 1
00100 => [3,3] => 100100 => 011011 => 0
00101 => [3,2,1] => 100101 => 011010 => 1
00110 => [3,1,2] => 100110 => 011001 => 1
00111 => [3,1,1,1] => 100111 => 011000 => 0
01000 => [2,4] => 101000 => 010111 => 2
01001 => [2,3,1] => 101001 => 010110 => 1
01010 => [2,2,2] => 101010 => 010101 => 3
01011 => [2,2,1,1] => 101011 => 010100 => 2
01100 => [2,1,3] => 101100 => 010011 => 1
01101 => [2,1,2,1] => 101101 => 010010 => 2
01110 => [2,1,1,2] => 101110 => 010001 => 2
01111 => [2,1,1,1,1] => 101111 => 010000 => 1
10000 => [1,5] => 110000 => 001111 => 0
10001 => [1,4,1] => 110001 => 001110 => 1
10010 => [1,3,2] => 110010 => 001101 => 1
10011 => [1,3,1,1] => 110011 => 001100 => 0
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St001230
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
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]
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001465
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,1] => 1
1 => [1,1] => [1,0,1,0]
 => [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.