Identifier
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.