Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001365
(load all 3 compositions to match this statistic)
Mapping
St001365: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 2
1 => 1
00 => 4
01 => 3
10 => 2
11 => 1
000 => 8
001 => 7
010 => 6
011 => 4
100 => 4
101 => 3
110 => 2
111 => 1
0000 => 16
0001 => 15
0010 => 14
0011 => 11
0100 => 12
0101 => 10
0110 => 8
0111 => 5
1000 => 8
1001 => 7
1010 => 6
1011 => 4
1100 => 4
1101 => 3
1110 => 2
1111 => 1
00000 => 32
00001 => 31
00010 => 30
00011 => 26
00100 => 28
00101 => 25
00110 => 22
00111 => 16
01000 => 24
01001 => 22
01010 => 20
01011 => 15
01100 => 16
01101 => 13
01110 => 10
01111 => 6
10000 => 16
10001 => 15
10010 => 14
10011 => 11
Description
The number of lattice paths of the same length weakly above the path given by a binary word. In particular, there are $2^n$ lattice paths weakly above the the length $n$ binary word $0\dots 0$, there is a unique path weakly above $1\dots 1$, and there are $\binom{2n}{n}$ paths weakly above the length $2n$ binary word $10\dots 10$.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 3%●distinct values known / distinct values provided: 2%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8} - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {4,6,7,8} - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {4,6,7,8} - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {4,6,7,8} - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => 2 = 3 - 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]
 => ? ∊ {4,6,7,8,8,10,11,12,14,15,16} - 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]
 => 3 = 4 - 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]
 => 4 = 5 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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 = 2 - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,13,14,15,15,16,16,16,20,22,22,24,25,26,28,30,31,32} - 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 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]
 => ? ∊ {4,6,7,8,8,10,10,11,12,12,13,14,15,15,16,16,16,16,19,20,20,21,22,22,22,24,25,26,26,28,29,30,30,31,32,32,32,35,37,38,40,41,42,44,44,46,48,50,52,53,56,56,57,60,62,63,64} - 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.