Processing math: 100%

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 [w1,w2] in P is small if w2 is obtained from w1 by replacing some valleys with peaks.
This statistic counts the number of chains w=w1<<wd=11 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 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 (k1)-bounded partition [1, Theorem 7], see also [2, Section 1.2].
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.