Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001361
(load all 2 compositions to match this statistic)
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: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
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 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? ∊ {32,48,56,60,62,64,80,80,88,88,92,96,100,104,104,106,112,116,120,122,128,128,140,144,148,150,152,152,152,162,164,164,168,180,182,192,194,204,212,222,224,252}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => ? ∊ {64,96,112,120,124,126,128,160,160,176,176,184,188,192,200,208,208,212,218,224,224,228,232,232,238,240,244,250,256,256,280,288,296,300,304,304,304,304,310,312,324,328,328,332,336,336,338,340,344,350,352,360,364,364,368,374,376,376,382,384,388,392,396,400,402,408,416,420,424,432,436,440,444,444,448,448,462,462,464,472,472,476,480,492,494,504,504,512,524,524,528,532,540,544,550,556,560,560,560,562,568,568,574,576,580,584,592,602,612,616,616,618,632,644,644,672,672,672,680,682,720,724,724,724,760,770,784,784,824,840,840,924}
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 $10\dots 10$.