Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000389
(load all 3 compositions to match this statistic)
Mapping
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 1
00 => 0
01 => 1
10 => 1
11 => 0
000 => 0
001 => 1
010 => 1
011 => 0
100 => 1
101 => 2
110 => 0
111 => 1
0000 => 0
0001 => 1
0010 => 1
0011 => 0
0100 => 1
0101 => 2
0110 => 0
0111 => 1
1000 => 1
1001 => 2
1010 => 2
1011 => 1
1100 => 0
1101 => 1
1110 => 1
1111 => 0
00000 => 0
00001 => 1
00010 => 1
00011 => 0
00100 => 1
00101 => 2
00110 => 0
00111 => 1
01000 => 1
01001 => 2
01010 => 2
01011 => 1
01100 => 0
01101 => 1
01110 => 1
01111 => 0
10000 => 1
10001 => 2
10010 => 2
10011 => 1
Description
The number of runs of ones of odd length in a binary word.
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
(load all 2 compositions to match this statistic)
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
(load all 2 compositions to match this statistic)
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: St001878
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
0 => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,1}
1 => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,1}
00 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
001 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
010 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
011 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
100 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
101 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
110 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
111 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
0000 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0001 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0011 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0111 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1000 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1100 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1110 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1111 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
00000 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00001 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00011 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
00111 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01000 => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01100 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01110 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01111 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10000 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10011 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
000110 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
000111 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
001100 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
001110 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
011000 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
011100 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
100011 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
100111 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
110001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
110011 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
111000 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
111001 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
0000110 => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 1
0000111 => [4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 1
0001000 => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
0001100 => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ([(0,2),(2,1)],3)
 => 1
0001110 => [3,3,1] => [[5,5,3],[4,2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
0001111 => [3,4] => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
0010001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
0010011 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
0011000 => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
0011001 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 1
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
0011011 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
0011100 => [2,3,2] => [[5,4,2],[3,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
0011110 => [2,4,1] => [[5,5,2],[4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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: 16% ●values known / values provided: 16%●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
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,5,4,3,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,4,3,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [5,4,3,2,1,7,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [5,4,3,2,1,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [4,3,2,1,7,6,5,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [4,3,2,1,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [4,3,2,1,5,8,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [4,3,2,1,5,7,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [4,3,2,1,5,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,2,1,7,6,5,4,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,6,5,4,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,6,5,4,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [3,2,1,5,4,8,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,7,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [3,2,1,5,4,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [3,2,1,5,4,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [3,2,1,4,8,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,2,1,4,7,6,5,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [3,2,1,4,6,5,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [3,2,1,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,5,7,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,7,6,5,4,3,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,6,5,4,3,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3,8,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
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: 14% ●values known / values provided: 14%●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
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,5,4,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,4,3,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,4,3,2,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,4,3,2,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,4,3,2,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,4,3,2,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,5,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001877
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
01 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1}
10 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
11 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
000 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
001 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
010 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2}
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
100 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2}
110 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
0000 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0001 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0010 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0100 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1000 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1100 => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
00000 => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00001 => [5,1] => [[5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00010 => [4,2] => [[5,4],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00100 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01000 => [2,4] => [[5,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10000 => [1,5] => [[5,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10100 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11000 => [1,1,4] => [[4,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
000000 => [7] => [[7],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000001 => [6,1] => [[6,6],[5]]
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000100 => [4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 0
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 0
001000 => [3,4] => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ([(0,2),(2,1)],3)
 => 0
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010010 => [2,3,2] => [[5,4,2],[3,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010100 => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 2
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
100010 => [1,4,2] => [[5,4,1],[3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
100100 => [1,3,3] => [[5,3,1],[2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 2
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
101000 => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 2
101010 => [1,2,2,2] => [[4,3,2,1],[2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 2
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
110010 => [1,1,3,2] => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,2),(2,1)],3)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001876
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
01 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1}
10 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
11 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
000 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
001 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
010 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2}
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
100 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2}
110 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2}
0000 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0001 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0010 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0100 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1000 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1100 => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2}
00000 => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00001 => [5,1] => [[5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00010 => [4,2] => [[5,4],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00100 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01000 => [2,4] => [[5,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10000 => [1,5] => [[5,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
10100 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11000 => [1,1,4] => [[4,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
000000 => [7] => [[7],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000001 => [6,1] => [[6,6],[5]]
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000100 => [4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 0
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 0
001000 => [3,4] => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ([(0,2),(2,1)],3)
 => 0
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010010 => [2,3,2] => [[5,4,2],[3,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
010100 => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
100010 => [1,4,2] => [[5,4,1],[3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
100100 => [1,3,3] => [[5,3,1],[2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
101000 => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
110010 => [1,1,3,2] => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,2),(2,1)],3)
 => 0
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
111011 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
0001000 => [4,4] => [[7,4],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
0001001 => [4,3,1] => [[6,6,4],[5,3]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
0001010 => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => ([(0,2),(2,1)],3)
 => 0
0001100 => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 0
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
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: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
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
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
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St000455The second largest eigenvalue of a graph if it is integral.