Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001361
Mapping
St001361: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2
[1,0,1,0]
 => 6
[1,1,0,0]
 => 4
[1,0,1,0,1,0]
 => 20
[1,0,1,1,0,0]
 => 16
[1,1,0,0,1,0]
 => 14
[1,1,0,1,0,0]
 => 12
[1,1,1,0,0,0]
 => 8
[1,0,1,0,1,0,1,0]
 => 70
[1,0,1,0,1,1,0,0]
 => 60
[1,0,1,1,0,0,1,0]
 => 58
[1,0,1,1,0,1,0,0]
 => 52
[1,0,1,1,1,0,0,0]
 => 40
[1,1,0,0,1,0,1,0]
 => 50
[1,1,0,0,1,1,0,0]
 => 44
[1,1,0,1,0,0,1,0]
 => 44
[1,1,0,1,0,1,0,0]
 => 40
[1,1,0,1,1,0,0,0]
 => 32
[1,1,1,0,0,0,1,0]
 => 30
[1,1,1,0,0,1,0,0]
 => 28
[1,1,1,0,1,0,0,0]
 => 24
[1,1,1,1,0,0,0,0]
 => 16
[1,0,1,0,1,0,1,0,1,0]
 => 252
[1,0,1,0,1,0,1,1,0,0]
 => 224
[1,0,1,0,1,1,0,0,1,0]
 => 222
[1,0,1,0,1,1,0,1,0,0]
 => 204
[1,0,1,0,1,1,1,0,0,0]
 => 168
[1,0,1,1,0,0,1,0,1,0]
 => 212
[1,0,1,1,0,0,1,1,0,0]
 => 192
[1,0,1,1,0,1,0,0,1,0]
 => 194
[1,0,1,1,0,1,0,1,0,0]
 => 180
[1,0,1,1,0,1,1,0,0,0]
 => 152
[1,0,1,1,1,0,0,0,1,0]
 => 152
[1,0,1,1,1,0,0,1,0,0]
 => 144
[1,0,1,1,1,0,1,0,0,0]
 => 128
[1,0,1,1,1,1,0,0,0,0]
 => 96
[1,1,0,0,1,0,1,0,1,0]
 => 182
[1,1,0,0,1,0,1,1,0,0]
 => 164
[1,1,0,0,1,1,0,0,1,0]
 => 164
[1,1,0,0,1,1,0,1,0,0]
 => 152
[1,1,0,0,1,1,1,0,0,0]
 => 128
[1,1,0,1,0,0,1,0,1,0]
 => 162
[1,1,0,1,0,0,1,1,0,0]
 => 148
[1,1,0,1,0,1,0,0,1,0]
 => 150
[1,1,0,1,0,1,0,1,0,0]
 => 140
[1,1,0,1,0,1,1,0,0,0]
 => 120
[1,1,0,1,1,0,0,0,1,0]
 => 122
[1,1,0,1,1,0,0,1,0,0]
 => 116
[1,1,0,1,1,0,1,0,0,0]
 => 104
[1,1,0,1,1,1,0,0,0,0]
 => 80
Description
The number of lattice paths of the same length that stay weakly above a Dyck path. In particular, the statistic value is $2^n$ for the Dyck path consisting of $n$ north steps followed by $n$ east steps and the central binomial coefficient $\binom{2n}{n}$ for the Dyck path consisting of $n$ alternating north and east steps. The number of such paths is always even: the final step of a Dyck path $D$ must be a down step. Thus, the final step of a path above $D$ can be arbitrarily chosen.
Matching statistic: St001365
(load all 2 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
St001365: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 2%
Values
[1,0]
 => 10 => 2
[1,0,1,0]
 => 1010 => 6
[1,1,0,0]
 => 1100 => 4
[1,0,1,0,1,0]
 => 101010 => 20
[1,0,1,1,0,0]
 => 101100 => 16
[1,1,0,0,1,0]
 => 110010 => 14
[1,1,0,1,0,0]
 => 110100 => 12
[1,1,1,0,0,0]
 => 111000 => 8
[1,0,1,0,1,0,1,0]
 => 10101010 => 70
[1,0,1,0,1,1,0,0]
 => 10101100 => 60
[1,0,1,1,0,0,1,0]
 => 10110010 => 58
[1,0,1,1,0,1,0,0]
 => 10110100 => 52
[1,0,1,1,1,0,0,0]
 => 10111000 => 40
[1,1,0,0,1,0,1,0]
 => 11001010 => 50
[1,1,0,0,1,1,0,0]
 => 11001100 => 44
[1,1,0,1,0,0,1,0]
 => 11010010 => 44
[1,1,0,1,0,1,0,0]
 => 11010100 => 40
[1,1,0,1,1,0,0,0]
 => 11011000 => 32
[1,1,1,0,0,0,1,0]
 => 11100010 => 30
[1,1,1,0,0,1,0,0]
 => 11100100 => 28
[1,1,1,0,1,0,0,0]
 => 11101000 => 24
[1,1,1,1,0,0,0,0]
 => 11110000 => 16
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 252
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 224
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 222
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 204
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 168
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? = 212
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 192
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 194
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 180
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 152
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? = 152
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 144
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 128
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 96
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? = 182
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? = 164
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? = 164
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? = 152
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 128
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? = 162
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? = 148
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? = 150
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? = 140
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? = 120
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 122
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? = 116
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? = 104
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? = 80
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? = 112
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 104
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 106
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 100
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? = 88
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 92
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 88
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? = 80
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 64
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 62
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 60
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? = 56
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? = 48
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? = 32
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 924
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 840
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => ? = 840
[1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 784
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 672
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => ? = 824
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? = 760
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => ? = 770
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 $10\dots 10$.