Identifier
Values
([2],3) => [2] => [1,1,0,0,1,0] => 110010 => 1
([1,1],3) => [1,1] => [1,0,1,1,0,0] => 101100 => 1
([3,1],3) => [2,1] => [1,0,1,0,1,0] => 101010 => 1
([2],4) => [2] => [1,1,0,0,1,0] => 110010 => 1
([1,1],4) => [1,1] => [1,0,1,1,0,0] => 101100 => 1
([2,1],4) => [2,1] => [1,0,1,0,1,0] => 101010 => 1
([2],5) => [2] => [1,1,0,0,1,0] => 110010 => 1
([1,1],5) => [1,1] => [1,0,1,1,0,0] => 101100 => 1
([2,1],5) => [2,1] => [1,0,1,0,1,0] => 101010 => 1
([2],6) => [2] => [1,1,0,0,1,0] => 110010 => 1
([1,1],6) => [1,1] => [1,0,1,1,0,0] => 101100 => 1
([2,1],6) => [2,1] => [1,0,1,0,1,0] => 101010 => 1
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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
to bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.