Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000518
(load all 5 compositions to match this statistic)
Mapping
St000518: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 2
1 => 2
00 => 3
01 => 4
10 => 4
11 => 3
000 => 4
001 => 6
010 => 7
011 => 6
100 => 6
101 => 7
110 => 6
111 => 4
0000 => 5
0001 => 8
0010 => 10
0011 => 9
0100 => 10
0101 => 12
0110 => 11
0111 => 8
1000 => 8
1001 => 11
1010 => 12
1011 => 10
1100 => 9
1101 => 10
1110 => 8
1111 => 5
00000 => 6
00001 => 10
00010 => 13
00011 => 12
00100 => 14
00101 => 17
00110 => 16
00111 => 12
01000 => 13
01001 => 18
01010 => 20
01011 => 17
01100 => 16
01101 => 18
01110 => 15
01111 => 10
10000 => 10
10001 => 15
10010 => 18
10011 => 16
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: 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: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 19%
Values
0 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 3 = 2 + 1
1 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => 3 = 2 + 1
00 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
01 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 5 = 4 + 1
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5 = 4 + 1
11 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
010 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 1
110 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
0000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
0001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 8 + 1
0010 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 11 = 10 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 10 = 9 + 1
0100 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 11 = 10 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 13 = 12 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 12 = 11 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 8 + 1
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9 = 8 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 12 = 11 + 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 13 = 12 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 11 = 10 + 1
1100 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10 = 9 + 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 11 = 10 + 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9 = 8 + 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
00000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 6 + 1
00001 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 11 = 10 + 1
00010 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => 14 = 13 + 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 13 = 12 + 1
00100 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => 15 = 14 + 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 18 = 17 + 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => 17 = 16 + 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 13 = 12 + 1
01000 => [2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => 14 = 13 + 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 19 = 18 + 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 21 = 20 + 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 18 = 17 + 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => 17 = 16 + 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 19 = 18 + 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => 16 = 15 + 1
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 = 10 + 1
10000 => [1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11 = 10 + 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => 16 = 15 + 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => 19 = 18 + 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => 17 = 16 + 1
0000001 => [7,1] => [[7,7],[6]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 14 + 1
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 19 + 1
0000011 => [6,1,1] => [[6,6,6],[5,5]]
 => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 18 + 1
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 22 + 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 27 + 1
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 26 + 1
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 20 + 1
0001000 => [4,4] => [[7,4],[3]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
 => ? = 23 + 1
0001001 => [4,3,1] => [[6,6,4],[5,3]]
 => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
 => ? = 32 + 1
0001010 => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 36 + 1
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
 => ? = 31 + 1
0001100 => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
 => ? = 30 + 1
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 34 + 1
0001110 => [4,1,1,2] => [[5,4,4,4],[3,3,3]]
 => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
 => ? = 29 + 1
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 20 + 1
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 22 + 1
0010001 => [3,4,1] => [[6,6,3],[5,2]]
 => ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
 => ? = 33 + 1
0010010 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
 => ? = 40 + 1
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]]
 => ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
 => ? = 36 + 1
0010100 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
 => ? = 39 + 1
0010101 => [3,2,2,1] => [[5,5,4,3],[4,3,2]]
 => ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
 => ? = 46 + 1
0010110 => [3,2,1,2] => [[5,4,4,3],[3,3,2]]
 => ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
 => ? = 42 + 1
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
 => ? = 31 + 1
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 30 + 1
0011001 => [3,1,3,1] => [[5,5,3,3],[4,2,2]]
 => ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
 => ? = 40 + 1
0011010 => [3,1,2,2] => [[5,4,3,3],[3,2,2]]
 => ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
 => ? = 43 + 1
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
 => ? = 36 + 1
0011100 => [3,1,1,3] => [[5,3,3,3],[2,2,2]]
 => ([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
 => ? = 32 + 1
0011101 => [3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
 => ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 35 + 1
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 28 + 1
0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 18 + 1
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 19 + 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 30 + 1
0100010 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 38 + 1
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]]
 => ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 35 + 1
0100100 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 40 + 1
0100101 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 48 + 1
0100110 => [2,3,1,2] => [[5,4,4,2],[3,3,1]]
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
 => ? = 45 + 1
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 34 + 1
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 36 + 1
0101001 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
 => ? = 49 + 1
0101010 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
 => ? = 54 + 1
0101011 => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
 => ? = 46 + 1
0101100 => [2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
 => ? = 43 + 1
0101101 => [2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 48 + 1
0101110 => [2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]]
 => ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
 => ? = 40 + 1
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
 => ? = 27 + 1
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? = 26 + 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 38 + 1
0110010 => [2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
 => ? = 45 + 1
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: 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: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 19%
Values
0 => [2] => [[2],[]]
 => 3 = 2 + 1
1 => [1,1] => [[1,1],[]]
 => 3 = 2 + 1
00 => [3] => [[3],[]]
 => 4 = 3 + 1
01 => [2,1] => [[2,2],[1]]
 => 5 = 4 + 1
10 => [1,2] => [[2,1],[]]
 => 5 = 4 + 1
11 => [1,1,1] => [[1,1,1],[]]
 => 4 = 3 + 1
000 => [4] => [[4],[]]
 => 5 = 4 + 1
001 => [3,1] => [[3,3],[2]]
 => 7 = 6 + 1
010 => [2,2] => [[3,2],[1]]
 => 8 = 7 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => 7 = 6 + 1
100 => [1,3] => [[3,1],[]]
 => 7 = 6 + 1
101 => [1,2,1] => [[2,2,1],[1]]
 => 8 = 7 + 1
110 => [1,1,2] => [[2,1,1],[]]
 => 7 = 6 + 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => 5 = 4 + 1
0000 => [5] => [[5],[]]
 => 6 = 5 + 1
0001 => [4,1] => [[4,4],[3]]
 => 9 = 8 + 1
0010 => [3,2] => [[4,3],[2]]
 => 11 = 10 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => 10 = 9 + 1
0100 => [2,3] => [[4,2],[1]]
 => 11 = 10 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => 13 = 12 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => 12 = 11 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => 9 = 8 + 1
1000 => [1,4] => [[4,1],[]]
 => 9 = 8 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => 12 = 11 + 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => 13 = 12 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 11 = 10 + 1
1100 => [1,1,3] => [[3,1,1],[]]
 => 10 = 9 + 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => 11 = 10 + 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => 9 = 8 + 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => 6 = 5 + 1
00000 => [6] => [[6],[]]
 => 7 = 6 + 1
00001 => [5,1] => [[5,5],[4]]
 => 11 = 10 + 1
00010 => [4,2] => [[5,4],[3]]
 => 14 = 13 + 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => 13 = 12 + 1
00100 => [3,3] => [[5,3],[2]]
 => 15 = 14 + 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => 18 = 17 + 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => 17 = 16 + 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => 13 = 12 + 1
01000 => [2,4] => [[5,2],[1]]
 => 14 = 13 + 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => 19 = 18 + 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => 21 = 20 + 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => 18 = 17 + 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => 17 = 16 + 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => 19 = 18 + 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => 16 = 15 + 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => 11 = 10 + 1
10000 => [1,5] => [[5,1],[]]
 => 11 = 10 + 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => 16 = 15 + 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => 19 = 18 + 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => 17 = 16 + 1
0000000 => [8] => [[8],[]]
 => ? = 8 + 1
0000001 => [7,1] => [[7,7],[6]]
 => ? = 14 + 1
0000010 => [6,2] => [[7,6],[5]]
 => ? = 19 + 1
0000011 => [6,1,1] => [[6,6,6],[5,5]]
 => ? = 18 + 1
0000100 => [5,3] => [[7,5],[4]]
 => ? = 22 + 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ? = 27 + 1
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ? = 26 + 1
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => ? = 20 + 1
0001000 => [4,4] => [[7,4],[3]]
 => ? = 23 + 1
0001001 => [4,3,1] => [[6,6,4],[5,3]]
 => ? = 32 + 1
0001010 => [4,2,2] => [[6,5,4],[4,3]]
 => ? = 36 + 1
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => ? = 31 + 1
0001100 => [4,1,3] => [[6,4,4],[3,3]]
 => ? = 30 + 1
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ? = 34 + 1
0001110 => [4,1,1,2] => [[5,4,4,4],[3,3,3]]
 => ? = 29 + 1
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => ? = 20 + 1
0010000 => [3,5] => [[7,3],[2]]
 => ? = 22 + 1
0010001 => [3,4,1] => [[6,6,3],[5,2]]
 => ? = 33 + 1
0010010 => [3,3,2] => [[6,5,3],[4,2]]
 => ? = 40 + 1
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]]
 => ? = 36 + 1
0010100 => [3,2,3] => [[6,4,3],[3,2]]
 => ? = 39 + 1
0010101 => [3,2,2,1] => [[5,5,4,3],[4,3,2]]
 => ? = 46 + 1
0010110 => [3,2,1,2] => [[5,4,4,3],[3,3,2]]
 => ? = 42 + 1
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ? = 31 + 1
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ? = 30 + 1
0011001 => [3,1,3,1] => [[5,5,3,3],[4,2,2]]
 => ? = 40 + 1
0011010 => [3,1,2,2] => [[5,4,3,3],[3,2,2]]
 => ? = 43 + 1
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => ? = 36 + 1
0011100 => [3,1,1,3] => [[5,3,3,3],[2,2,2]]
 => ? = 32 + 1
0011101 => [3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
 => ? = 35 + 1
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ? = 28 + 1
0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => ? = 18 + 1
0100000 => [2,6] => [[7,2],[1]]
 => ? = 19 + 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ? = 30 + 1
0100010 => [2,4,2] => [[6,5,2],[4,1]]
 => ? = 38 + 1
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]]
 => ? = 35 + 1
0100100 => [2,3,3] => [[6,4,2],[3,1]]
 => ? = 40 + 1
0100101 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ? = 48 + 1
0100110 => [2,3,1,2] => [[5,4,4,2],[3,3,1]]
 => ? = 45 + 1
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ? = 34 + 1
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ? = 36 + 1
0101001 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ? = 49 + 1
0101010 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ? = 54 + 1
0101011 => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => ? = 46 + 1
0101100 => [2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => ? = 43 + 1
0101101 => [2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]]
 => ? = 48 + 1
0101110 => [2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]]
 => ? = 40 + 1
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ? = 27 + 1
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ? = 26 + 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ? = 38 + 1
Description
The total number of rook placements on a Ferrers board.
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: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Values
0 => [2] => ([],2)
 => 2
1 => [1,1] => ([(0,1)],2)
 => 2
00 => [3] => ([],3)
 => 3
01 => [2,1] => ([(0,2),(1,2)],3)
 => 4
10 => [1,2] => ([(1,2)],3)
 => 4
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
000 => [4] => ([],4)
 => 4
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 6
010 => [2,2] => ([(1,3),(2,3)],4)
 => 7
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
100 => [1,3] => ([(2,3)],4)
 => 6
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 7
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 6
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0000 => [5] => ([],5)
 => 5
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 8
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 10
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 10
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 12
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 11
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
1000 => [1,4] => ([(3,4)],5)
 => 8
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 11
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 12
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 9
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
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
00000 => [6] => ([],6)
 => 6
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 13
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 14
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 17
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 16
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12
01000 => [2,4] => ([(3,5),(4,5)],6)
 => 13
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 20
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 17
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 16
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 15
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
10000 => [1,5] => ([(4,5)],6)
 => 10
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 15
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 18
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 16
000000 => [7] => ([],7)
 => ? = 7
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 12
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 16
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 15
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 18
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 21
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 18
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 25
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 28
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 24
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 23
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 26
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
 => ? = 15
010000 => [2,5] => ([(4,6),(5,6)],7)
 => ? = 16
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 24
010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 29
010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 26
010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 28
010101 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 33
010110 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 30
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
 => ? = 22
011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 21
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 28
011010 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 30
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(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)
 => ? = 25
011100 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
011101 => [2,1,1,2,1] => ([(0,6),(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)
 => ? = 24
011110 => [2,1,1,1,2] => ([(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)
 => ? = 19
011111 => [2,1,1,1,1,1] => ([(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)
 => ? = 12
100000 => [1,6] => ([(5,6)],7)
 => ? = 12
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 19
100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 24
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 25
100101 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 30
100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 28
100111 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
 => ? = 21
101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 30
101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 33
101011 => [1,2,2,1,1] => ([(0,5),(0,6),(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)
 => ? = 28
101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 26
101101 => [1,2,1,2,1] => ([(0,6),(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)
 => ? = 29
101110 => [1,2,1,1,2] => ([(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)
 => ? = 24
101111 => [1,2,1,1,1,1] => ([(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)
 => ? = 16
110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ? = 15
110001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 22
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: 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: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 4 = 3 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 4 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 4 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5 = 4 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 7 = 6 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 8 = 7 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 7 = 6 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 7 = 6 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 8 = 7 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 7 = 6 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5 = 4 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6 = 5 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 9 = 8 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 11 = 10 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 10 = 9 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 11 = 10 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 13 = 12 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 12 = 11 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 9 = 8 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 9 = 8 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 12 = 11 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 13 = 12 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 11 = 10 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 10 = 9 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 11 = 10 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 9 = 8 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6 = 5 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 7 = 6 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 11 = 10 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => 14 = 13 + 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 = 12 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 14 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 17 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 16 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 12 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 13 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 18 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 20 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 17 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 16 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 18 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 15 + 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 = 10 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 10 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 15 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 18 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 16 + 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 17 + 1
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 20 + 1
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 18 + 1
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 13 + 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 12 + 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 16 + 1
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 17 + 1
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 14 + 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 = 12 + 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 = 13 + 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 = 10 + 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 = 6 + 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 = 7 + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 12 + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => ? = 16 + 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ? = 15 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [2,3,4,6,1,7,8,5] => ? = 18 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [2,3,4,6,1,8,5,7] => ? = 22 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [2,3,4,7,1,5,8,6] => ? = 21 + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,8,1,5,6,7] => ? = 16 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [2,3,5,1,6,7,8,4] => ? = 18 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [2,3,5,1,6,8,4,7] => ? = 25 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,3,5,1,7,4,8,6] => ? = 28 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [2,3,5,1,8,4,6,7] => ? = 24 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [2,3,6,1,4,7,8,5] => ? = 23 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [2,3,6,1,4,8,5,7] => ? = 26 + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [2,3,7,1,4,5,8,6] => ? = 22 + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ? = 15 + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,1,5,6,7,8,3] => ? = 16 + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,4,1,5,6,8,3,7] => ? = 24 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,4,1,5,7,3,8,6] => ? = 29 + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,4,1,5,8,3,6,7] => ? = 26 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,6,3,7,8,5] => ? = 28 + 1
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,8,5,7] => ? = 33 + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,4,1,7,3,5,8,6] => ? = 30 + 1
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,4,1,8,3,5,6,7] => ? = 22 + 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,5,1,3,6,7,8,4] => ? = 21 + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,5,1,3,6,8,4,7] => ? = 28 + 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,7,4,8,6] => ? = 30 + 1
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,5,1,3,8,4,6,7] => ? = 25 + 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 = 7 + 1
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: 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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 7%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 4 = 3 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 4 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 4 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5 = 4 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 7 = 6 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 8 = 7 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 7 = 6 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 7 = 6 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 8 = 7 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 7 = 6 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5 = 4 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6 = 5 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 9 = 8 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 11 = 10 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 10 = 9 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 11 = 10 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 13 = 12 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 12 = 11 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 9 = 8 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 9 = 8 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 12 = 11 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 13 = 12 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 11 = 10 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 10 = 9 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 11 = 10 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 9 = 8 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6 = 5 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 6 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 10 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 13 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 12 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 14 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 17 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 16 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 12 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 13 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 18 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 20 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 17 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 16 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 18 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 15 + 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] => ? = 10 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 10 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 15 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 18 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 16 + 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 17 + 1
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 20 + 1
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 18 + 1
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 13 + 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 12 + 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 16 + 1
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 17 + 1
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 14 + 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] => ? = 12 + 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] => ? = 13 + 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] => ? = 10 + 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] => ? = 6 + 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] => ? = 7 + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 12 + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => ? = 16 + 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ? = 15 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [2,3,4,6,1,7,8,5] => ? = 18 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [2,3,4,6,1,8,5,7] => ? = 22 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [2,3,4,7,1,5,8,6] => ? = 21 + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,8,1,5,6,7] => ? = 16 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [2,3,5,1,6,7,8,4] => ? = 18 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [2,3,5,1,6,8,4,7] => ? = 25 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,3,5,1,7,4,8,6] => ? = 28 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [2,3,5,1,8,4,6,7] => ? = 24 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [2,3,6,1,4,7,8,5] => ? = 23 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [2,3,6,1,4,8,5,7] => ? = 26 + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [2,3,7,1,4,5,8,6] => ? = 22 + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ? = 15 + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,1,5,6,7,8,3] => ? = 16 + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,4,1,5,6,8,3,7] => ? = 24 + 1
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.