Identifier
Values
[1] => 10 => 00 => 00 => 1
[2] => 100 => 000 => 000 => 1
[1,1] => 110 => 010 => 100 => 1
[3] => 1000 => 0000 => 0000 => 1
[2,1] => 1010 => 0010 => 0100 => 2
[1,1,1] => 1110 => 0110 => 1001 => 1
[4] => 10000 => 00000 => 00000 => 1
[3,1] => 10010 => 00010 => 00100 => 3
[2,2] => 1100 => 0100 => 1000 => 1
[2,1,1] => 10110 => 00110 => 01001 => 2
[1,1,1,1] => 11110 => 01110 => 10011 => 1
[5] => 100000 => 000000 => 000000 => 1
[4,1] => 100010 => 000010 => 000100 => 4
[3,2] => 10100 => 00100 => 01000 => 3
[3,1,1] => 100110 => 000110 => 001001 => 3
[2,2,1] => 11010 => 01010 => 10100 => 2
[2,1,1,1] => 101110 => 001110 => 010011 => 2
[1,1,1,1,1] => 111110 => 011110 => 100111 => 1
[4,2] => 100100 => 000100 => 001000 => 6
[3,3] => 11000 => 01000 => 10000 => 1
[3,2,1] => 101010 => 001010 => 010100 => 5
[2,2,2] => 11100 => 01100 => 10001 => 1
[2,2,1,1] => 110110 => 010110 => 101001 => 2
[4,3] => 101000 => 001000 => 010000 => 4
[3,3,1] => 110010 => 010010 => 100100 => 3
[3,2,2] => 101100 => 001100 => 010001 => 3
[2,2,2,1] => 111010 => 011010 => 101100 => 1
[4,4] => 110000 => 010000 => 100000 => 1
[3,3,2] => 110100 => 010100 => 101000 => 3
[2,2,2,2] => 111100 => 011100 => 100011 => 1
[3,3,3] => 111000 => 011000 => 100001 => 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 partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
rowmotion
Description
Return the rowmotion of the binary word, regarded as an order ideal in the product of two chains.
In particular, this operation preserves the number of ones, and its order is the length of the word, see section 3.3 of [1].
Map
twist
Description
Return the binary word with first letter inverted.