Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤ
Values
[1,1] => [1,0,1,0] => [1] => 10 => 0
[1,1,1] => [1,0,1,0,1,0] => [2,1] => 1010 => 0
[1,2] => [1,0,1,1,0,0] => [1,1] => 110 => 1
[2,1] => [1,1,0,0,1,0] => [2] => 100 => -1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 101010 => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 11010 => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 100110 => 0
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 10100 => -1
[2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 1100 => 0
[3,1] => [1,1,1,0,0,0,1,0] => [3] => 1000 => -2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 10101010 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 1101010 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 10011010 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 111010 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10100110 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 1100110 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 10001110 => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 1010100 => -1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 110100 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1001100 => -1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 101000 => -2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => -1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => -3
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1010101010 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 110101010 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 1001101010 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 11101010 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 1010011010 => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 110011010 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 1000111010 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1111010 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 1010100110 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 110100110 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 1001100110 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 11100110 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 1010001110 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 110001110 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 1000011110 => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 101010100 => -1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 11010100 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 100110100 => -1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 1110100 => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 101001100 => -1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 11001100 => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 100011100 => -1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 111100 => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 10101000 => -2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 1101000 => -1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 10011000 => -2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 111000 => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 1010000 => -3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 110000 => -2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 100000 => -4
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 101010101010 => 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2,1] => 111101010 => 3
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 11111010 => 4
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3,1,1] => 111100110 => 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1111110 => 5
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [5,5,4,3,2] => 1101010100 => 0
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [4,4,4,3,2] => 111010100 => 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [5,5,3,3,2] => 1100110100 => 0
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2] => 11110100 => 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [5,5,4,2,2] => 1101001100 => 0
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2] => 111001100 => 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 1111100 => 3
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [5,5,4,3] => 110101000 => -1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3] => 11101000 => 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [5,5,3,3] => 110011000 => -1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => 1111000 => 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [6,5,4] => 101010000 => -3
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [5,5,4] => 11010000 => -2
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [6,4,4] => 100110000 => -3
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,4,4] => 1110000 => -1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => 10100000 => -4
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 1100000 => -3
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 1000000 => -5
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => 111111010 => 5
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 11111110 => 6
[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [6,6,5,4,3,2] => 110101010100 => 0
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3,2] => 111110100 => 3
[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2] => 11111100 => 4
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3] => 11111000 => 2
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [4,4,4,4] => 11110000 => 0
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [6,6,5] => 110100000 => -3
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [5,5,5] => 11100000 => -2
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [7,6] => 101000000 => -5
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [6,6] => 11000000 => -4
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 10000000 => -6
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 111111110 => 7
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2] => 111111100 => 5
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,3] => 111111000 => 3
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [4,4,4,4,4] => 111110000 => 1
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [5,5,5,5] => 111100000 => -1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [6,6,6] => 111000000 => -3
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [7,7] => 110000000 => -5
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 100000000 => -7
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => 1111111110 => 8
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searching the database for the individual values of this statistic
Description
The number of ones minus the number of zeros of a binary word.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
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