Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001658
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => 3
1 => [1,1] => [[1,1],[]]
 => 3
00 => [3] => [[3],[]]
 => 4
01 => [2,1] => [[2,2],[1]]
 => 5
10 => [1,2] => [[2,1],[]]
 => 5
11 => [1,1,1] => [[1,1,1],[]]
 => 4
000 => [4] => [[4],[]]
 => 5
001 => [3,1] => [[3,3],[2]]
 => 7
010 => [2,2] => [[3,2],[1]]
 => 8
011 => [2,1,1] => [[2,2,2],[1,1]]
 => 7
100 => [1,3] => [[3,1],[]]
 => 7
101 => [1,2,1] => [[2,2,1],[1]]
 => 8
110 => [1,1,2] => [[2,1,1],[]]
 => 7
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => 5
0000 => [5] => [[5],[]]
 => 6
0001 => [4,1] => [[4,4],[3]]
 => 9
0010 => [3,2] => [[4,3],[2]]
 => 11
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => 10
0100 => [2,3] => [[4,2],[1]]
 => 11
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => 13
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => 12
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => 9
1000 => [1,4] => [[4,1],[]]
 => 9
1001 => [1,3,1] => [[3,3,1],[2]]
 => 12
1010 => [1,2,2] => [[3,2,1],[1]]
 => 13
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 11
1100 => [1,1,3] => [[3,1,1],[]]
 => 10
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => 11
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => 9
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => 6
00000 => [6] => [[6],[]]
 => 7
00001 => [5,1] => [[5,5],[4]]
 => 11
00010 => [4,2] => [[5,4],[3]]
 => 14
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => 13
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => 11
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => 13
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => 14
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => 11
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => 7
000000 => [7] => [[7],[]]
 => 8
111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => 8
=> [1] => [[1],[]]
 => 2
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000070
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 3
1 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => 3
00 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 4
01 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 5
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5
11 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4
000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 7
010 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 8
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 7
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 8
110 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
0000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
0001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9
0010 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 11
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 10
0100 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 11
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 13
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 12
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 12
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 13
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 11
1100 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 11
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
00000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
00001 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 11
00010 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => 14
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 13
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 11
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => 14
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
000000 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 8
111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 8
=> [1] => [[1],[]]
 => ([],1)
 => 2
Description
The number of antichains in a poset. An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable. An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000110
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 3
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 3
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 4
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 5
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 5
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 4
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 7
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 8
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 7
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 7
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 8
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 7
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 9
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 11
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 10
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 11
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 13
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 12
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 9
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 12
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 13
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 11
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 10
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 11
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 9
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 7
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 11
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => 14
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => 13
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => 11
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => 13
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => 14
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => 11
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 7
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 8
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => 8
=> [1] => [1,0]
 => [2,1] => 2
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000518
(load all 5 compositions to match this statistic)
Mapping
St000518: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Values
0 => 2 = 3 - 1
1 => 2 = 3 - 1
00 => 3 = 4 - 1
01 => 4 = 5 - 1
10 => 4 = 5 - 1
11 => 3 = 4 - 1
000 => 4 = 5 - 1
001 => 6 = 7 - 1
010 => 7 = 8 - 1
011 => 6 = 7 - 1
100 => 6 = 7 - 1
101 => 7 = 8 - 1
110 => 6 = 7 - 1
111 => 4 = 5 - 1
0000 => 5 = 6 - 1
0001 => 8 = 9 - 1
0010 => 10 = 11 - 1
0011 => 9 = 10 - 1
0100 => 10 = 11 - 1
0101 => 12 = 13 - 1
0110 => 11 = 12 - 1
0111 => 8 = 9 - 1
1000 => 8 = 9 - 1
1001 => 11 = 12 - 1
1010 => 12 = 13 - 1
1011 => 10 = 11 - 1
1100 => 9 = 10 - 1
1101 => 10 = 11 - 1
1110 => 8 = 9 - 1
1111 => 5 = 6 - 1
00000 => 6 = 7 - 1
00001 => 10 = 11 - 1
00010 => 13 = 14 - 1
00011 => 12 = 13 - 1
01111 => 10 = 11 - 1
11100 => 12 = 13 - 1
11101 => 13 = 14 - 1
11110 => 10 = 11 - 1
11111 => 6 = 7 - 1
000000 => 7 = 8 - 1
111111 => 7 = 8 - 1
=> ? = 2 - 1
Description
The number of distinct subsequences in a binary word. In contrast to the subword complexity [[St000294]] this is the cardinality of the set of all subsequences of not necessarily consecutive letters.
Matching statistic: St000087
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 2 = 3 - 1
1 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
00 => [3] => ([],3)
 => 3 = 4 - 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 5 - 1
10 => [1,2] => ([(1,2)],3)
 => 4 = 5 - 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 4 - 1
000 => [4] => ([],4)
 => 4 = 5 - 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 6 = 7 - 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => 7 = 8 - 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6 = 7 - 1
100 => [1,3] => ([(2,3)],4)
 => 6 = 7 - 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 7 = 8 - 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 6 = 7 - 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
0000 => [5] => ([],5)
 => 5 = 6 - 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 8 = 9 - 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 10 = 11 - 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9 = 10 - 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 10 = 11 - 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 12 = 13 - 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 11 = 12 - 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8 = 9 - 1
1000 => [1,4] => ([(3,4)],5)
 => 8 = 9 - 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 11 = 12 - 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 12 = 13 - 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10 = 11 - 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 9 = 10 - 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10 = 11 - 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8 = 9 - 1
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
00000 => [6] => ([],6)
 => 6 = 7 - 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10 = 11 - 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 13 = 14 - 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12 = 13 - 1
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 10 = 11 - 1
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12 = 13 - 1
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 13 = 14 - 1
11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 10 = 11 - 1
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 7 - 1
000000 => [7] => ([],7)
 => ? = 8 - 1
111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
=> [1] => ([],1)
 => 1 = 2 - 1
Description
The number of induced subgraphs. A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.
Matching statistic: St001464
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 92%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 3
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 3
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 4
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 5
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 5
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 4
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 7
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 8
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 7
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 7
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 8
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 7
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 9
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 11
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 10
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 11
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 13
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 12
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 9
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 12
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 13
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 11
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 10
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 11
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 9
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 7
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 11
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 13
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 11
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 13
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 14
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 11
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 7
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ? = 8
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? = 8
=> [1] => [1,0]
 => [2,1] => 2
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000656
(load all 3 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
St000656: Posets ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 46%
Values
0 => ([(0,1)],2)
 => 2 = 3 - 1
1 => ([(0,1)],2)
 => 2 = 3 - 1
00 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
11 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
000 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 7 = 8 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 7 = 8 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
111 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 10 - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 13 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 12 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 12 - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 13 - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 10 - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 14 - 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 13 - 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 13 - 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 14 - 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 8 - 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 8 - 1
=> ?
 => ? = 2 - 1
Description
The number of cuts of a poset. A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St001880
(load all 4 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
St001880: Posets ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 38%
Values
0 => ([(0,1)],2)
 => ? = 3 - 1
1 => ([(0,1)],2)
 => ? = 3 - 1
00 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
11 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
000 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 8 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 8 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6 = 7 - 1
111 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 10 - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 13 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 12 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 12 - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 13 - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 10 - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 11 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 9 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 14 - 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 13 - 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 13 - 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 14 - 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 11 - 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
=> ?
 => ? = 2 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001684
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 46%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 2 = 3 - 1
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 4 = 5 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4 = 5 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 6 = 7 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 7 = 8 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 6 = 7 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 6 = 7 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 7 = 8 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 6 = 7 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4 = 5 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 6 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 9 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 11 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ? = 10 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 11 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 13 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 12 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ? = 9 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 9 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 12 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 13 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 11 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 10 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 11 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 9 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 6 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 7 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 11 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 13 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 11 - 1
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 13 - 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 14 - 1
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 11 - 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 7 - 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ? = 8 - 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? = 8 - 1
=> [1] => [1,0]
 => [2,1] => 1 = 2 - 1
Description
The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].
Matching statistic: St001875
(load all 3 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 38%
Values
0 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
1 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
00 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5
11 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
000 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 7
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 8
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 7
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 7
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 8
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 7
111 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 9
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 11
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 10
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 11
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(0,8),(2,11),(3,11),(4,9),(5,9),(6,10),(7,10),(8,2),(8,3),(9,1),(10,4),(10,5),(11,6),(11,7)],12)
 => ? = 13
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ([(0,9),(2,17),(3,11),(3,15),(4,11),(4,14),(5,12),(6,13),(7,10),(8,7),(8,17),(9,2),(9,8),(10,14),(10,15),(11,18),(12,16),(13,16),(14,5),(14,18),(15,6),(15,18),(16,1),(17,3),(17,4),(17,10),(18,12),(18,13)],19)
 => ? = 12
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 9
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 9
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ([(0,9),(2,17),(3,11),(3,15),(4,11),(4,14),(5,12),(6,13),(7,10),(8,7),(8,17),(9,2),(9,8),(10,14),(10,15),(11,18),(12,16),(13,16),(14,5),(14,18),(15,6),(15,18),(16,1),(17,3),(17,4),(17,10),(18,12),(18,13)],19)
 => ? = 12
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(0,8),(2,11),(3,11),(4,9),(5,9),(6,10),(7,10),(8,2),(8,3),(9,1),(10,4),(10,5),(11,6),(11,7)],12)
 => ? = 13
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 11
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 10
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 11
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 9
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ?
 => ? = 11
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ?
 => ? = 14
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ?
 => ? = 13
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ?
 => ? = 11
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ?
 => ? = 13
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ?
 => ? = 14
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ?
 => ? = 11
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
=> ?
 => ?
 => ? = 2
Description
The number of simple modules with projective dimension at most 1.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St001060The distinguishing index of a graph.