- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

[1] => [1,0] => [1] => 1 => 1

[1,2] => [1,0,1,0] => [1,1] => 11 => 2

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

[1,2,3] => [1,0,1,0,1,0] => [1,1,1] => 111 => 3

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

[2,1,3] => [1,1,0,0,1,0] => [2,1] => 101 => 2

[2,3,1] => [1,1,0,1,0,0] => [2,1] => 101 => 2

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

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

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 1111 => 4

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

[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,2,1] => 1101 => 3

[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,2,1] => 1101 => 3

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

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

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,1,1] => 1011 => 3

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

[2,3,1,4] => [1,1,0,1,0,0,1,0] => [2,1,1] => 1011 => 3

[2,3,4,1] => [1,1,0,1,0,1,0,0] => [2,1,1] => 1011 => 3

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

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

[3,1,2,4] => [1,1,1,0,0,0,1,0] => [3,1] => 1001 => 2

[3,1,4,2] => [1,1,1,0,0,1,0,0] => [3,1] => 1001 => 2

[3,2,1,4] => [1,1,1,0,0,0,1,0] => [3,1] => 1001 => 2

[3,2,4,1] => [1,1,1,0,0,1,0,0] => [3,1] => 1001 => 2

[3,4,1,2] => [1,1,1,0,1,0,0,0] => [3,1] => 1001 => 2

[3,4,2,1] => [1,1,1,0,1,0,0,0] => [3,1] => 1001 => 2

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

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

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

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

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

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

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 11111 => 5

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

[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 11101 => 4

[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => 11101 => 4

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

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

[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 11011 => 4

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

[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => 11011 => 4

[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => 11011 => 4

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

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

[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11001 => 3

[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,3,1] => 11001 => 3

[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11001 => 3

[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,3,1] => 11001 => 3

[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,3,1] => 11001 => 3

[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,3,1] => 11001 => 3

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

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

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

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

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

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

[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 10111 => 4

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

[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 10101 => 3

[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [2,2,1] => 10101 => 3

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

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

[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => 10111 => 4

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

[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => 10111 => 4

[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => 10111 => 4

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

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

[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [2,2,1] => 10101 => 3

[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [2,2,1] => 10101 => 3

[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [2,2,1] => 10101 => 3

[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [2,2,1] => 10101 => 3

[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [2,2,1] => 10101 => 3

[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [2,2,1] => 10101 => 3

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

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

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

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

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

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

[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 10011 => 3

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

[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [3,1,1] => 10011 => 3

[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [3,1,1] => 10011 => 3

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

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

[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 10011 => 3

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

[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [3,1,1] => 10011 => 3

[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,1] => 10011 => 3

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

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

[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [3,1,1] => 10011 => 3

[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [3,1,1] => 10011 => 3

[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [3,1,1] => 10011 => 3

[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [3,1,1] => 10011 => 3

[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [3,1,1] => 10011 => 3

[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [3,1,1] => 10011 => 3

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

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + 3\ q^{2} + q^{3}$

$F_{4} = 6\ q + 11\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = 24\ q + 50\ q^{2} + 35\ q^{3} + 10\ q^{4} + q^{5}$

$F_{6} = 120\ q + 274\ q^{2} + 225\ q^{3} + 85\ q^{4} + 15\ q^{5} + q^{6}$

$F_{7} = 720\ q + 1764\ q^{2} + 1624\ q^{3} + 735\ q^{4} + 175\ q^{5} + 21\ q^{6} + q^{7}$

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.

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 binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer

$i$ as the word

$10\dots 0$ consisting of a one followed by

$i-1$ zeros, the binary word of a composition

$(i_1,\dots,i_k)$ is the concatenation of of words for

$i_1,\dots,i_k$ .
The image of this map contains precisely the words which do not begin with a

$0$ .