Identifier
-
Mp00077:
Semistandard tableaux
—shape⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤ
Values
[[1,2]] => [2] => [1,0,1,0] => 1010 => 1
[[2,2]] => [2] => [1,0,1,0] => 1010 => 1
[[1],[2]] => [1,1] => [1,1,0,0] => 1100 => 1
[[1,3]] => [2] => [1,0,1,0] => 1010 => 1
[[2,3]] => [2] => [1,0,1,0] => 1010 => 1
[[3,3]] => [2] => [1,0,1,0] => 1010 => 1
[[1],[3]] => [1,1] => [1,1,0,0] => 1100 => 1
[[2],[3]] => [1,1] => [1,1,0,0] => 1100 => 1
[[1,1,2]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,2,2]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,2,2]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,1],[2]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,2],[2]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,4]] => [2] => [1,0,1,0] => 1010 => 1
[[2,4]] => [2] => [1,0,1,0] => 1010 => 1
[[3,4]] => [2] => [1,0,1,0] => 1010 => 1
[[4,4]] => [2] => [1,0,1,0] => 1010 => 1
[[1],[4]] => [1,1] => [1,1,0,0] => 1100 => 1
[[2],[4]] => [1,1] => [1,1,0,0] => 1100 => 1
[[3],[4]] => [1,1] => [1,1,0,0] => 1100 => 1
[[1,1,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,2,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,3,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,2,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,3,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,3,3]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,1],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,2],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,3],[2]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,3],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,2],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,3],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1],[2],[3]] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 2
[[1,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[1,5]] => [2] => [1,0,1,0] => 1010 => 1
[[2,5]] => [2] => [1,0,1,0] => 1010 => 1
[[3,5]] => [2] => [1,0,1,0] => 1010 => 1
[[4,5]] => [2] => [1,0,1,0] => 1010 => 1
[[5,5]] => [2] => [1,0,1,0] => 1010 => 1
[[1],[5]] => [1,1] => [1,1,0,0] => 1100 => 1
[[2],[5]] => [1,1] => [1,1,0,0] => 1100 => 1
[[3],[5]] => [1,1] => [1,1,0,0] => 1100 => 1
[[4],[5]] => [1,1] => [1,1,0,0] => 1100 => 1
[[1,1,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,2,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,3,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,4,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,2,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,3,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,4,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,3,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,4,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[4,4,4]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,1],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,2],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,4],[2]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,3],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,4],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,4],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,2],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,3],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,4],[3]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[2,4],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[3,3],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[3,4],[4]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1],[2],[4]] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 2
[[1],[3],[4]] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 2
[[2],[3],[4]] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 2
[[1,1],[2,3]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[1,1],[3,3]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[1,2],[2,3]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[1,2],[3,3]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[2,2],[3,3]] => [2,2] => [1,1,1,0,0,0] => 111000 => 1
[[1,6]] => [2] => [1,0,1,0] => 1010 => 1
[[2,6]] => [2] => [1,0,1,0] => 1010 => 1
[[3,6]] => [2] => [1,0,1,0] => 1010 => 1
[[4,6]] => [2] => [1,0,1,0] => 1010 => 1
[[5,6]] => [2] => [1,0,1,0] => 1010 => 1
[[6,6]] => [2] => [1,0,1,0] => 1010 => 1
[[1],[6]] => [1,1] => [1,1,0,0] => 1100 => 1
[[2],[6]] => [1,1] => [1,1,0,0] => 1100 => 1
[[3],[6]] => [1,1] => [1,1,0,0] => 1100 => 1
[[4],[6]] => [1,1] => [1,1,0,0] => 1100 => 1
[[5],[6]] => [1,1] => [1,1,0,0] => 1100 => 1
[[1,1,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,2,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,3,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,4,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,5,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,2,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,3,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,4,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[2,5,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,3,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,4,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[3,5,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[4,4,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[4,5,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[5,5,5]] => [3] => [1,0,1,0,1,0] => 101010 => 1
[[1,1],[5]] => [2,1] => [1,0,1,1,0,0] => 101100 => 1
[[1,2],[5]] => [2,1] => [1,0,1,1,0,0] => 101100 => 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=1…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.
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=1…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
shape
Description
Return the shape of a tableau.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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