- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001722: Binary words ⟶ ℤ

Values

[[1]] => [1] => 1 => 1

[[1],[1]] => [1,1] => 11 => 1

[[2]] => [2] => 0 => 1

[[1,1]] => [2] => 0 => 1

[[1],[1],[1]] => [1,1,1] => 111 => 1

[[2],[1]] => [2,1] => 01 => 1

[[1,1],[1]] => [2,1] => 01 => 1

[[3]] => [3] => 1 => 1

[[2,1]] => [3] => 1 => 1

[[1,1,1]] => [3] => 1 => 1

[[1],[1],[1],[1]] => [1,1,1,1] => 1111 => 1

[[2],[1],[1]] => [2,1,1] => 011 => 1

[[2],[2]] => [2,2] => 00 => 1

[[1,1],[1],[1]] => [2,1,1] => 011 => 1

[[1,1],[1,1]] => [2,2] => 00 => 1

[[3],[1]] => [3,1] => 11 => 1

[[2,1],[1]] => [3,1] => 11 => 1

[[1,1,1],[1]] => [3,1] => 11 => 1

[[4]] => [4] => 0 => 1

[[3,1]] => [4] => 0 => 1

[[2,2]] => [4] => 0 => 1

[[2,1,1]] => [4] => 0 => 1

[[1,1,1,1]] => [4] => 0 => 1

[[1],[1],[1],[1],[1]] => [1,1,1,1,1] => 11111 => 1

[[2],[1],[1],[1]] => [2,1,1,1] => 0111 => 1

[[2],[2],[1]] => [2,2,1] => 001 => 1

[[1,1],[1],[1],[1]] => [2,1,1,1] => 0111 => 1

[[1,1],[1,1],[1]] => [2,2,1] => 001 => 1

[[3],[1],[1]] => [3,1,1] => 111 => 1

[[3],[2]] => [3,2] => 10 => 1

[[2,1],[1],[1]] => [3,1,1] => 111 => 1

[[2,1],[2]] => [3,2] => 10 => 1

[[2,1],[1,1]] => [3,2] => 10 => 1

[[1,1,1],[1],[1]] => [3,1,1] => 111 => 1

[[1,1,1],[1,1]] => [3,2] => 10 => 1

[[4],[1]] => [4,1] => 01 => 1

[[3,1],[1]] => [4,1] => 01 => 1

[[2,2],[1]] => [4,1] => 01 => 1

[[2,1,1],[1]] => [4,1] => 01 => 1

[[1,1,1,1],[1]] => [4,1] => 01 => 1

[[5]] => [5] => 1 => 1

[[4,1]] => [5] => 1 => 1

[[3,2]] => [5] => 1 => 1

[[3,1,1]] => [5] => 1 => 1

[[2,2,1]] => [5] => 1 => 1

[[2,1,1,1]] => [5] => 1 => 1

[[1,1,1,1,1]] => [5] => 1 => 1

[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => 111111 => 1

[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => 01111 => 1

[[2],[2],[1],[1]] => [2,2,1,1] => 0011 => 1

[[2],[2],[2]] => [2,2,2] => 000 => 1

[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => 01111 => 1

[[1,1],[1,1],[1],[1]] => [2,2,1,1] => 0011 => 1

[[1,1],[1,1],[1,1]] => [2,2,2] => 000 => 1

[[3],[1],[1],[1]] => [3,1,1,1] => 1111 => 1

[[3],[2],[1]] => [3,2,1] => 101 => 1

[[3],[3]] => [3,3] => 11 => 1

[[2,1],[1],[1],[1]] => [3,1,1,1] => 1111 => 1

[[2,1],[2],[1]] => [3,2,1] => 101 => 1

[[2,1],[1,1],[1]] => [3,2,1] => 101 => 1

[[2,1],[2,1]] => [3,3] => 11 => 1

[[1,1,1],[1],[1],[1]] => [3,1,1,1] => 1111 => 1

[[1,1,1],[1,1],[1]] => [3,2,1] => 101 => 1

[[1,1,1],[1,1,1]] => [3,3] => 11 => 1

[[4],[1],[1]] => [4,1,1] => 011 => 1

[[4],[2]] => [4,2] => 00 => 1

[[3,1],[1],[1]] => [4,1,1] => 011 => 1

[[3,1],[2]] => [4,2] => 00 => 1

[[3,1],[1,1]] => [4,2] => 00 => 1

[[2,2],[1],[1]] => [4,1,1] => 011 => 1

[[2,2],[2]] => [4,2] => 00 => 1

[[2,2],[1,1]] => [4,2] => 00 => 1

[[2,1,1],[1],[1]] => [4,1,1] => 011 => 1

[[2,1,1],[2]] => [4,2] => 00 => 1

[[2,1,1],[1,1]] => [4,2] => 00 => 1

[[1,1,1,1],[1],[1]] => [4,1,1] => 011 => 1

[[1,1,1,1],[1,1]] => [4,2] => 00 => 1

[[5],[1]] => [5,1] => 11 => 1

[[4,1],[1]] => [5,1] => 11 => 1

[[3,2],[1]] => [5,1] => 11 => 1

[[3,1,1],[1]] => [5,1] => 11 => 1

[[2,2,1],[1]] => [5,1] => 11 => 1

[[2,1,1,1],[1]] => [5,1] => 11 => 1

[[1,1,1,1,1],[1]] => [5,1] => 11 => 1

[[6]] => [6] => 0 => 1

[[5,1]] => [6] => 0 => 1

[[4,2]] => [6] => 0 => 1

[[4,1,1]] => [6] => 0 => 1

[[3,3]] => [6] => 0 => 1

[[3,2,1]] => [6] => 0 => 1

[[3,1,1,1]] => [6] => 0 => 1

[[2,2,2]] => [6] => 0 => 1

[[2,2,1,1]] => [6] => 0 => 1

[[2,1,1,1,1]] => [6] => 0 => 1

[[1,1,1,1,1,1]] => [6] => 0 => 1

[[2],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 011111 => 1

[[2],[2],[1],[1],[1]] => [2,2,1,1,1] => 00111 => 1

[[2],[2],[2],[1]] => [2,2,2,1] => 0001 => 1

[[1,1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 011111 => 1

[[1,1],[1,1],[1],[1],[1]] => [2,2,1,1,1] => 00111 => 1

[[1,1],[1,1],[1,1],[1]] => [2,2,2,1] => 0001 => 1

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$F_{1} = q$

$F_{2} = 3\ q$

$F_{3} = 6\ q$

$F_{4} = 13\ q$

$F_{5} = 24\ q$

$F_{6} = 48\ q$

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 partition

Description

The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.

Map

odd parts

Description

Return the binary word indicating which parts of the partition are odd.