- FindStat

Identifier

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤ

Values

[1,1] => [1] => [1,0,1,0] => 1010 => 1
[2,1] => [1] => [1,0,1,0] => 1010 => 1
[1,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[3,1] => [1] => [1,0,1,0] => 1010 => 1
[2,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[2,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[4,1] => [1] => [1,0,1,0] => 1010 => 1
[3,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[3,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[2,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[5,1] => [1] => [1,0,1,0] => 1010 => 1
[4,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[4,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[3,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[6,1] => [1] => [1,0,1,0] => 1010 => 1
[5,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[5,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[4,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[7,1] => [1] => [1,0,1,0] => 1010 => 1
[6,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[6,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[5,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[8,1] => [1] => [1,0,1,0] => 1010 => 1
[7,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[7,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[6,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[9,1] => [1] => [1,0,1,0] => 1010 => 1
[8,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[8,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[7,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[10,1] => [1] => [1,0,1,0] => 1010 => 1
[9,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[9,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[8,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[11,1] => [1] => [1,0,1,0] => 1010 => 1
[10,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[10,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[9,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[12,1] => [1] => [1,0,1,0] => 1010 => 1
[11,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[11,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[10,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[13,1] => [1] => [1,0,1,0] => 1010 => 1
[12,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[12,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[11,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[14,1] => [1] => [1,0,1,0] => 1010 => 1
[13,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[13,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[12,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[15,1] => [1] => [1,0,1,0] => 1010 => 1
[14,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[14,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[13,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[16,1] => [1] => [1,0,1,0] => 1010 => 1
[15,2] => [2] => [1,1,0,0,1,0] => 110010 => 1
[15,1,1] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[14,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 1

search for individual values

searching the database for the individual values of this statistic

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

first row removal

Description

Removes the first entry of an integer partition

Map

to binary word

Description

Return the Dyck word as binary word.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.