- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [4,1] => 10001 => 2

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

[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,1] => 10001 => 2

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

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

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

[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [4,1] => 10001 => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,1,2,1,1] => 111011 => 5

[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [2,2,1,1] => 101011 => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,3,1,1] => 110011 => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [1,3,1,1] => 110011 => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [4,1,1] => 100011 => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,3,1,1] => 110011 => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [4,1,1] => 100011 => 3

[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,2,2,1] => 110101 => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [1,2,2,1] => 110101 => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [3,2,1] => 100101 => 3

[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,4,1] => 110001 => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [1,4,1] => 110001 => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [1,4,1] => 110001 => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [5,1] => 100001 => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,1,3,1] => 111001 => 4

[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [2,3,1] => 101001 => 3

[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,4,1] => 110001 => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [1,4,1] => 110001 => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [5,1] => 100001 => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,2,2,1] => 110101 => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [3,2,1] => 100101 => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,2,2,1] => 110101 => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [1,2,1,2] => 110110 => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [3,1,2] => 100110 => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,4,1] => 110001 => 3

[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [1,3,2] => 110010 => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [1,3,2] => 110010 => 3

[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [4,2] => 100010 => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,4,1] => 110001 => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [5,1] => 100001 => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,4,1] => 110001 => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [1,3,2] => 110010 => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [4,2] => 100010 => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,4,1] => 110001 => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [1,3,2] => 110010 => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [1,2,3] => 110100 => 3

[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [3,3] => 100100 => 2

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Description

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

Map

DEX composition

Description

The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.

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 $0$.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].