Identifier
Values
{{1}} => [1] => [1,0] => 10 => 1
{{1,2}} => [2,1] => [1,1,0,0] => 1100 => 1
{{1},{2}} => [1,2] => [1,0,1,0] => 1010 => 1
{{1,2,3}} => [2,3,1] => [1,1,0,1,0,0] => 110100 => 2
{{1,2},{3}} => [2,1,3] => [1,1,0,0,1,0] => 110010 => 1
{{1,3},{2}} => [3,2,1] => [1,1,1,0,0,0] => 111000 => 1
{{1},{2,3}} => [1,3,2] => [1,0,1,1,0,0] => 101100 => 1
{{1},{2},{3}} => [1,2,3] => [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 permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.