- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤ

Values

[1,2] => [1,0,1,0] => [1] => 10 => 0

[1,2,3] => [1,0,1,0,1,0] => [2,1] => 1010 => 0

[1,3,2] => [1,0,1,1,0,0] => [1,1] => 110 => 0

[2,1,3] => [1,1,0,0,1,0] => [2] => 100 => 0

[2,3,1] => [1,1,0,1,0,0] => [1] => 10 => 0

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [3,2,1] => 101010 => 0

[1,2,4,3] => [1,0,1,0,1,1,0,0] => [2,2,1] => 11010 => 0

[1,3,2,4] => [1,0,1,1,0,0,1,0] => [3,1,1] => 100110 => 0

[1,3,4,2] => [1,0,1,1,0,1,0,0] => [2,1,1] => 10110 => 0

[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => 0

[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => 0

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [3,2] => 10100 => 0

[2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,2] => 1100 => 0

[2,3,1,4] => [1,1,0,1,0,0,1,0] => [3,1] => 10010 => 0

[2,3,4,1] => [1,1,0,1,0,1,0,0] => [2,1] => 1010 => 0

[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1] => 110 => 0

[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,1] => 110 => 0

[3,1,2,4] => [1,1,1,0,0,0,1,0] => [3] => 1000 => 0

[3,1,4,2] => [1,1,1,0,0,1,0,0] => [2] => 100 => 0

[3,2,1,4] => [1,1,1,0,0,0,1,0] => [3] => 1000 => 0

[3,2,4,1] => [1,1,1,0,0,1,0,0] => [2] => 100 => 0

[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1] => 10 => 0

[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1] => 10 => 0

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 10101010 => 0

[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 1101010 => 0

[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 10011010 => 0

[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 1011010 => 0

[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 111010 => 0

[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 111010 => 0

[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10100110 => 0

[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 1100110 => 0

[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 10010110 => 0

[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 1010110 => 0

[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 110110 => 0

[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 110110 => 0

[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 10001110 => 0

[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 1001110 => 0

[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 10001110 => 0

[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 1001110 => 0

[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 101110 => 0

[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 101110 => 0

[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 0

[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 1010100 => 0

[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 110100 => 0

[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1001100 => 0

[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 101100 => 0

[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => 0

[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => 0

[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 1010010 => 0

[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 110010 => 0

[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 1001010 => 0

[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 101010 => 0

[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 11010 => 0

[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 11010 => 0

[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 1000110 => 0

[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 100110 => 0

[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 1000110 => 0

[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 100110 => 0

[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 10110 => 0

[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 10110 => 0

[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0

[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 101000 => 0

[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => 0

[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 100100 => 0

[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 10100 => 0

[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => 0

[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => 0

[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 101000 => 0

[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => 0

[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 100100 => 0

[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 10100 => 0

[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => 0

[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => 0

[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 100010 => 0

[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => 10010 => 0

[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 100010 => 0

[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => 10010 => 0

[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [2,1] => 1010 => 0

[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [2,1] => 1010 => 0

[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 0

[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => 0

[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [3] => 1000 => 0

[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => 0

[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [3] => 1000 => 0

[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [2] => 100 => 0

[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [2] => 100 => 0

>>> Load all 24466 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is palindromic.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

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

$(n-1,\ldots,2,1)$ .

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.