Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000764
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 1
010 => [1,1,1] => 1
011 => [1,2] => 2
100 => [1,2] => 2
101 => [1,1,1] => 1
110 => [2,1] => 1
111 => [3] => 1
0000 => [4] => 1
0001 => [3,1] => 1
0010 => [2,1,1] => 1
0011 => [2,2] => 1
0100 => [1,1,2] => 2
0101 => [1,1,1,1] => 1
0110 => [1,2,1] => 2
0111 => [1,3] => 2
1000 => [1,3] => 2
1001 => [1,2,1] => 2
1010 => [1,1,1,1] => 1
1011 => [1,1,2] => 2
1100 => [2,2] => 1
1101 => [2,1,1] => 1
1110 => [3,1] => 1
1111 => [4] => 1
00000 => [5] => 1
00001 => [4,1] => 1
00010 => [3,1,1] => 1
00011 => [3,2] => 1
00100 => [2,1,2] => 1
00101 => [2,1,1,1] => 1
00110 => [2,2,1] => 1
00111 => [2,3] => 2
01000 => [1,1,3] => 2
01001 => [1,1,2,1] => 2
01010 => [1,1,1,1,1] => 1
01011 => [1,1,1,2] => 2
01100 => [1,2,2] => 2
01101 => [1,2,1,1] => 2
01110 => [1,3,1] => 2
01111 => [1,4] => 2
10000 => [1,4] => 2
10001 => [1,3,1] => 2
10010 => [1,2,1,1] => 2
10011 => [1,2,2] => 2
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St001732
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Values
0 => [1] => [1,0]
 => 1
1 => [1] => [1,0]
 => 1
00 => [2] => [1,1,0,0]
 => 1
01 => [1,1] => [1,0,1,0]
 => 1
10 => [1,1] => [1,0,1,0]
 => 1
11 => [2] => [1,1,0,0]
 => 1
000 => [3] => [1,1,1,0,0,0]
 => 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1
111 => [3] => [1,1,1,0,0,0]
 => 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
00000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
00000011 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
00000101 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1
00000111 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1
00001111 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
00010101 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
00011110 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
00011111 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.