- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001722: Binary words ⟶ ℤ

Values

[1] => [1,0] => 10 => 01 => 1
[2] => [1,0,1,0] => 1010 => 0101 => 1
[1,1] => [1,1,0,0] => 1100 => 0011 => 1
[3] => [1,0,1,0,1,0] => 101010 => 010101 => 1
[2,1] => [1,0,1,1,0,0] => 101100 => 001101 => 3
[1,1,1] => [1,1,0,1,0,0] => 110100 => 001011 => 1
[2,2] => [1,1,1,0,0,0] => 111000 => 000111 => 1

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$F_{1} = q$

$F_{2} = 2\ q$

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

Description

The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence

$01$ , or a trailing

$0$ . A peak is a subsequence

$10$ or a trailing

$1$ . Let

$P$ be the lattice on binary words of length

$n$ , where the covering elements of a word are obtained by replacing a valley with a peak. An interval

$[w_1, w_2]$ in

$P$ is small if

$w_2$ is obtained from

$w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains

$w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word

$0110$ :

$0110 < 1011 < 1101 < 1110 < 1111$
and

$0110 < 1010 < 1101 < 1110 < 1111.$

Map

to binary word

Description

Return the Dyck word as binary word.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

reverse

Description

Return the reversal of a binary word.