Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 6 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 2 = 1 + 1
1 => 1 => [1,1] => 1 = 0 + 1
00 => 00 => [3] => 3 = 2 + 1
01 => 10 => [1,2] => 1 = 0 + 1
10 => 01 => [2,1] => 2 = 1 + 1
11 => 11 => [1,1,1] => 1 = 0 + 1
000 => 000 => [4] => 4 = 3 + 1
001 => 100 => [1,3] => 1 = 0 + 1
010 => 010 => [2,2] => 2 = 1 + 1
011 => 110 => [1,1,2] => 1 = 0 + 1
100 => 001 => [3,1] => 3 = 2 + 1
101 => 101 => [1,2,1] => 1 = 0 + 1
110 => 011 => [2,1,1] => 2 = 1 + 1
111 => 111 => [1,1,1,1] => 1 = 0 + 1
0000 => 0000 => [5] => 5 = 4 + 1
0001 => 1000 => [1,4] => 1 = 0 + 1
0010 => 0100 => [2,3] => 2 = 1 + 1
0011 => 1100 => [1,1,3] => 1 = 0 + 1
0100 => 0010 => [3,2] => 3 = 2 + 1
0101 => 1010 => [1,2,2] => 1 = 0 + 1
0110 => 0110 => [2,1,2] => 2 = 1 + 1
0111 => 1110 => [1,1,1,2] => 1 = 0 + 1
1000 => 0001 => [4,1] => 4 = 3 + 1
1001 => 1001 => [1,3,1] => 1 = 0 + 1
1010 => 0101 => [2,2,1] => 2 = 1 + 1
1011 => 1101 => [1,1,2,1] => 1 = 0 + 1
1100 => 0011 => [3,1,1] => 3 = 2 + 1
1101 => 1011 => [1,2,1,1] => 1 = 0 + 1
1110 => 0111 => [2,1,1,1] => 2 = 1 + 1
1111 => 1111 => [1,1,1,1,1] => 1 = 0 + 1
00000 => 00000 => [6] => 6 = 5 + 1
00001 => 10000 => [1,5] => 1 = 0 + 1
00010 => 01000 => [2,4] => 2 = 1 + 1
00011 => 11000 => [1,1,4] => 1 = 0 + 1
00100 => 00100 => [3,3] => 3 = 2 + 1
00101 => 10100 => [1,2,3] => 1 = 0 + 1
00110 => 01100 => [2,1,3] => 2 = 1 + 1
00111 => 11100 => [1,1,1,3] => 1 = 0 + 1
01000 => 00010 => [4,2] => 4 = 3 + 1
01001 => 10010 => [1,3,2] => 1 = 0 + 1
01010 => 01010 => [2,2,2] => 2 = 1 + 1
01011 => 11010 => [1,1,2,2] => 1 = 0 + 1
01100 => 00110 => [3,1,2] => 3 = 2 + 1
01101 => 10110 => [1,2,1,2] => 1 = 0 + 1
01110 => 01110 => [2,1,1,2] => 2 = 1 + 1
01111 => 11110 => [1,1,1,1,2] => 1 = 0 + 1
10000 => 00001 => [5,1] => 5 = 4 + 1
10001 => 10001 => [1,4,1] => 1 = 0 + 1
10010 => 01001 => [2,3,1] => 2 = 1 + 1
10011 => 11001 => [1,1,3,1] => 1 = 0 + 1
Description
The first part of an integer composition.
Matching statistic: St001176
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: 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] => ([(0,1)],2)
 => [2]
 => 0
00 => [3] => ([],3)
 => [1,1,1]
 => 2
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 0
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
000 => [4] => ([],4)
 => [1,1,1,1]
 => 3
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 4
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 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)
 => [5]
 => 0
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 3
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 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]
 => 0
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 5
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 0
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 2
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
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)
 => [6]
 => 0
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
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)
 => [6]
 => 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2
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)
 => [6]
 => 0
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 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)
 => [6]
 => 0
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 4
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
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)
 => [6]
 => 0
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => [1,1]
 => 2 = 1 + 1
1 => [1,1] => ([(0,1)],2)
 => [2]
 => 1 = 0 + 1
00 => [3] => ([],3)
 => [1,1,1]
 => 3 = 2 + 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
000 => [4] => ([],4)
 => [1,1,1,1]
 => 4 = 3 + 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 2 + 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 5 = 4 + 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 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)
 => [5]
 => 1 = 0 + 1
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4 = 3 + 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 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]
 => 1 = 0 + 1
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 6 = 5 + 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 2 + 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
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)
 => [6]
 => 1 = 0 + 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4 = 3 + 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
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)
 => [6]
 => 1 = 0 + 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 2 + 1
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)
 => [6]
 => 1 = 0 + 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 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)
 => [6]
 => 1 = 0 + 1
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 5 = 4 + 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
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)
 => [6]
 => 1 = 0 + 1
Description
The length of the partition.
Matching statistic: St000297
(load all 7 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => 11 => 2 = 1 + 1
1 => [1,1] => [2] => 10 => 1 = 0 + 1
00 => [3] => [1,1,1] => 111 => 3 = 2 + 1
01 => [2,1] => [2,1] => 101 => 1 = 0 + 1
10 => [1,2] => [1,2] => 110 => 2 = 1 + 1
11 => [1,1,1] => [3] => 100 => 1 = 0 + 1
000 => [4] => [1,1,1,1] => 1111 => 4 = 3 + 1
001 => [3,1] => [2,1,1] => 1011 => 1 = 0 + 1
010 => [2,2] => [1,2,1] => 1101 => 2 = 1 + 1
011 => [2,1,1] => [3,1] => 1001 => 1 = 0 + 1
100 => [1,3] => [1,1,2] => 1110 => 3 = 2 + 1
101 => [1,2,1] => [2,2] => 1010 => 1 = 0 + 1
110 => [1,1,2] => [1,3] => 1100 => 2 = 1 + 1
111 => [1,1,1,1] => [4] => 1000 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1] => 11111 => 5 = 4 + 1
0001 => [4,1] => [2,1,1,1] => 10111 => 1 = 0 + 1
0010 => [3,2] => [1,2,1,1] => 11011 => 2 = 1 + 1
0011 => [3,1,1] => [3,1,1] => 10011 => 1 = 0 + 1
0100 => [2,3] => [1,1,2,1] => 11101 => 3 = 2 + 1
0101 => [2,2,1] => [2,2,1] => 10101 => 1 = 0 + 1
0110 => [2,1,2] => [1,3,1] => 11001 => 2 = 1 + 1
0111 => [2,1,1,1] => [4,1] => 10001 => 1 = 0 + 1
1000 => [1,4] => [1,1,1,2] => 11110 => 4 = 3 + 1
1001 => [1,3,1] => [2,1,2] => 10110 => 1 = 0 + 1
1010 => [1,2,2] => [1,2,2] => 11010 => 2 = 1 + 1
1011 => [1,2,1,1] => [3,2] => 10010 => 1 = 0 + 1
1100 => [1,1,3] => [1,1,3] => 11100 => 3 = 2 + 1
1101 => [1,1,2,1] => [2,3] => 10100 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,4] => 11000 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [5] => 10000 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1] => 111111 => 6 = 5 + 1
00001 => [5,1] => [2,1,1,1,1] => 101111 => 1 = 0 + 1
00010 => [4,2] => [1,2,1,1,1] => 110111 => 2 = 1 + 1
00011 => [4,1,1] => [3,1,1,1] => 100111 => 1 = 0 + 1
00100 => [3,3] => [1,1,2,1,1] => 111011 => 3 = 2 + 1
00101 => [3,2,1] => [2,2,1,1] => 101011 => 1 = 0 + 1
00110 => [3,1,2] => [1,3,1,1] => 110011 => 2 = 1 + 1
00111 => [3,1,1,1] => [4,1,1] => 100011 => 1 = 0 + 1
01000 => [2,4] => [1,1,1,2,1] => 111101 => 4 = 3 + 1
01001 => [2,3,1] => [2,1,2,1] => 101101 => 1 = 0 + 1
01010 => [2,2,2] => [1,2,2,1] => 110101 => 2 = 1 + 1
01011 => [2,2,1,1] => [3,2,1] => 100101 => 1 = 0 + 1
01100 => [2,1,3] => [1,1,3,1] => 111001 => 3 = 2 + 1
01101 => [2,1,2,1] => [2,3,1] => 101001 => 1 = 0 + 1
01110 => [2,1,1,2] => [1,4,1] => 110001 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [5,1] => 100001 => 1 = 0 + 1
10000 => [1,5] => [1,1,1,1,2] => 111110 => 5 = 4 + 1
10001 => [1,4,1] => [2,1,1,2] => 101110 => 1 = 0 + 1
10010 => [1,3,2] => [1,2,1,2] => 110110 => 2 = 1 + 1
10011 => [1,3,1,1] => [3,1,2] => 100110 => 1 = 0 + 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 2 = 1 + 1
1 => [1,1] => 11 => 11 => 1 = 0 + 1
00 => [3] => 100 => 001 => 3 = 2 + 1
01 => [2,1] => 101 => 101 => 1 = 0 + 1
10 => [1,2] => 110 => 011 => 2 = 1 + 1
11 => [1,1,1] => 111 => 111 => 1 = 0 + 1
000 => [4] => 1000 => 0001 => 4 = 3 + 1
001 => [3,1] => 1001 => 1001 => 1 = 0 + 1
010 => [2,2] => 1010 => 0101 => 2 = 1 + 1
011 => [2,1,1] => 1011 => 1101 => 1 = 0 + 1
100 => [1,3] => 1100 => 0011 => 3 = 2 + 1
101 => [1,2,1] => 1101 => 1011 => 1 = 0 + 1
110 => [1,1,2] => 1110 => 0111 => 2 = 1 + 1
111 => [1,1,1,1] => 1111 => 1111 => 1 = 0 + 1
0000 => [5] => 10000 => 00001 => 5 = 4 + 1
0001 => [4,1] => 10001 => 10001 => 1 = 0 + 1
0010 => [3,2] => 10010 => 01001 => 2 = 1 + 1
0011 => [3,1,1] => 10011 => 11001 => 1 = 0 + 1
0100 => [2,3] => 10100 => 00101 => 3 = 2 + 1
0101 => [2,2,1] => 10101 => 10101 => 1 = 0 + 1
0110 => [2,1,2] => 10110 => 01101 => 2 = 1 + 1
0111 => [2,1,1,1] => 10111 => 11101 => 1 = 0 + 1
1000 => [1,4] => 11000 => 00011 => 4 = 3 + 1
1001 => [1,3,1] => 11001 => 10011 => 1 = 0 + 1
1010 => [1,2,2] => 11010 => 01011 => 2 = 1 + 1
1011 => [1,2,1,1] => 11011 => 11011 => 1 = 0 + 1
1100 => [1,1,3] => 11100 => 00111 => 3 = 2 + 1
1101 => [1,1,2,1] => 11101 => 10111 => 1 = 0 + 1
1110 => [1,1,1,2] => 11110 => 01111 => 2 = 1 + 1
1111 => [1,1,1,1,1] => 11111 => 11111 => 1 = 0 + 1
00000 => [6] => 100000 => 000001 => 6 = 5 + 1
00001 => [5,1] => 100001 => 100001 => 1 = 0 + 1
00010 => [4,2] => 100010 => 010001 => 2 = 1 + 1
00011 => [4,1,1] => 100011 => 110001 => 1 = 0 + 1
00100 => [3,3] => 100100 => 001001 => 3 = 2 + 1
00101 => [3,2,1] => 100101 => 101001 => 1 = 0 + 1
00110 => [3,1,2] => 100110 => 011001 => 2 = 1 + 1
00111 => [3,1,1,1] => 100111 => 111001 => 1 = 0 + 1
01000 => [2,4] => 101000 => 000101 => 4 = 3 + 1
01001 => [2,3,1] => 101001 => 100101 => 1 = 0 + 1
01010 => [2,2,2] => 101010 => 010101 => 2 = 1 + 1
01011 => [2,2,1,1] => 101011 => 110101 => 1 = 0 + 1
01100 => [2,1,3] => 101100 => 001101 => 3 = 2 + 1
01101 => [2,1,2,1] => 101101 => 101101 => 1 = 0 + 1
01110 => [2,1,1,2] => 101110 => 011101 => 2 = 1 + 1
01111 => [2,1,1,1,1] => 101111 => 111101 => 1 = 0 + 1
10000 => [1,5] => 110000 => 000011 => 5 = 4 + 1
10001 => [1,4,1] => 110001 => 100011 => 1 = 0 + 1
10010 => [1,3,2] => 110010 => 010011 => 2 = 1 + 1
10011 => [1,3,1,1] => 110011 => 110011 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => [1,1,0,0]
 => 3 = 1 + 2
1 => 1 => [1,1] => [1,0,1,0]
 => 2 = 0 + 2
00 => 00 => [3] => [1,1,1,0,0,0]
 => 4 = 2 + 2
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
11 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
000 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
011 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
110 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
111 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
0110 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
0111 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
1011 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
1101 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
1110 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 1 + 2
1111 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 5 + 2
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 1 + 2
00011 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4 = 2 + 2
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
00110 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
00111 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 5 = 3 + 2
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
01011 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
01100 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 2 + 2
01101 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 0 + 2
01110 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
01111 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 4 + 2
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 1 + 2
10011 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
0001100 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 2
0010100 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 2
0011100 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 2
0100100 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 2
0101100 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 2
0110100 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 2
1001100 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2 + 2
1010100 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 2
1100100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 2
1110000 => 0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000383
(load all 9 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 94%●distinct values known / distinct values provided: 88%
Values
0 => [2] => [2] => [2] => 2 = 1 + 1
1 => [1,1] => [1,1] => [1,1] => 1 = 0 + 1
00 => [3] => [3] => [3] => 3 = 2 + 1
01 => [2,1] => [2,1] => [2,1] => 1 = 0 + 1
10 => [1,2] => [1,2] => [1,2] => 2 = 1 + 1
11 => [1,1,1] => [1,1,1] => [1,1,1] => 1 = 0 + 1
000 => [4] => [4] => [4] => 4 = 3 + 1
001 => [3,1] => [3,1] => [3,1] => 1 = 0 + 1
010 => [2,2] => [2,2] => [2,2] => 2 = 1 + 1
011 => [2,1,1] => [1,2,1] => [2,1,1] => 1 = 0 + 1
100 => [1,3] => [1,3] => [1,3] => 3 = 2 + 1
101 => [1,2,1] => [2,1,1] => [1,2,1] => 1 = 0 + 1
110 => [1,1,2] => [1,1,2] => [1,1,2] => 2 = 1 + 1
111 => [1,1,1,1] => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
0000 => [5] => [5] => [5] => 5 = 4 + 1
0001 => [4,1] => [4,1] => [4,1] => 1 = 0 + 1
0010 => [3,2] => [3,2] => [3,2] => 2 = 1 + 1
0011 => [3,1,1] => [1,3,1] => [3,1,1] => 1 = 0 + 1
0100 => [2,3] => [2,3] => [2,3] => 3 = 2 + 1
0101 => [2,2,1] => [2,2,1] => [2,2,1] => 1 = 0 + 1
0110 => [2,1,2] => [2,1,2] => [2,1,2] => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,2,1] => [2,1,1,1] => 1 = 0 + 1
1000 => [1,4] => [1,4] => [1,4] => 4 = 3 + 1
1001 => [1,3,1] => [3,1,1] => [1,3,1] => 1 = 0 + 1
1010 => [1,2,2] => [1,2,2] => [1,2,2] => 2 = 1 + 1
1011 => [1,2,1,1] => [1,2,1,1] => [1,2,1,1] => 1 = 0 + 1
1100 => [1,1,3] => [1,1,3] => [1,1,3] => 3 = 2 + 1
1101 => [1,1,2,1] => [2,1,1,1] => [1,1,2,1] => 1 = 0 + 1
1110 => [1,1,1,2] => [1,1,1,2] => [1,1,1,2] => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1 = 0 + 1
00000 => [6] => [6] => [6] => 6 = 5 + 1
00001 => [5,1] => [5,1] => [5,1] => 1 = 0 + 1
00010 => [4,2] => [4,2] => [4,2] => 2 = 1 + 1
00011 => [4,1,1] => [1,4,1] => [4,1,1] => 1 = 0 + 1
00100 => [3,3] => [3,3] => [3,3] => 3 = 2 + 1
00101 => [3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
00110 => [3,1,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,3,1] => [3,1,1,1] => 1 = 0 + 1
01000 => [2,4] => [2,4] => [2,4] => 4 = 3 + 1
01001 => [2,3,1] => [2,3,1] => [2,3,1] => 1 = 0 + 1
01010 => [2,2,2] => [2,2,2] => [2,2,2] => 2 = 1 + 1
01011 => [2,2,1,1] => [1,2,2,1] => [2,1,2,1] => 1 = 0 + 1
01100 => [2,1,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
01101 => [2,1,2,1] => [2,2,1,1] => [1,2,2,1] => 1 = 0 + 1
01110 => [2,1,1,2] => [1,2,1,2] => [2,1,1,2] => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,1,2,1] => [2,1,1,1,1] => 1 = 0 + 1
10000 => [1,5] => [1,5] => [1,5] => 5 = 4 + 1
10001 => [1,4,1] => [4,1,1] => [1,4,1] => 1 = 0 + 1
10010 => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
10011 => [1,3,1,1] => [1,3,1,1] => [1,3,1,1] => 1 = 0 + 1
0000000 => [8] => [8] => [8] => ? = 7 + 1
0001010 => [4,2,2] => [2,4,2] => [4,2,2] => ? = 1 + 1
0001011 => [4,2,1,1] => [1,4,2,1] => [4,2,1,1] => ? = 0 + 1
0001100 => [4,1,3] => [1,4,3] => [4,1,3] => ? = 2 + 1
0001110 => [4,1,1,2] => [1,1,4,2] => [4,1,1,2] => ? = 1 + 1
0001111 => [4,1,1,1,1] => [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
0010111 => [3,2,1,1,1] => [1,1,3,2,1] => [3,2,1,1,1] => ? = 0 + 1
0011011 => [3,1,2,1,1] => [1,3,1,2,1] => [3,1,2,1,1] => ? = 0 + 1
0011101 => [3,1,1,2,1] => [3,1,1,2,1] => [3,1,1,2,1] => ? = 0 + 1
0011110 => [3,1,1,1,2] => [1,1,1,3,2] => [3,1,1,1,2] => ? = 1 + 1
1000111 => [1,4,1,1,1] => [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
1110000 => [1,1,1,5] => [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
1111110 => [1,1,1,1,1,1,2] => [1,1,1,1,1,1,2] => [1,1,1,1,1,1,2] => ? = 1 + 1
Description
The last part of an integer composition.
Matching statistic: St000678
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
1 => 0 => [2] => [1,1,0,0]
 => 1 = 0 + 1
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
11 => 00 => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
1100110 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0 + 1
1101110 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
1101111 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
1110010 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
1110101 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
1110111 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
1111101 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
1111110 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
1111111 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0 + 1
=> => [1] => [1,0]
 => ? = 0 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 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]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 1 + 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2 + 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1 + 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1 + 1
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1 + 1
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 1 + 1
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 1 + 1
1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 1 + 1
1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 1 + 1
1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 1 + 1
1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000617
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 75%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1 = 0 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1 = 0 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 1 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 0 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 0 + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 1 + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 0 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 0 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 1 + 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 1 + 1
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 0 + 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 2 + 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 1 + 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 0 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 1 + 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 0 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 0 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 1 + 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 0 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 0 + 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 1 + 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 0 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 0 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 0 + 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 0 + 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 0 + 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 2 + 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 0 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
Description
The number of global maxima of a Dyck path.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001733The number of weak left to right maxima of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000007The number of saliances of the permutation. St000989The number of final rises of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000260The radius of a connected graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000654The first descent of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000993The multiplicity of the largest part of an integer partition.