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Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths

Images

[] => []

[1] => [1,0]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]

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

[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0]

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]

[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]

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

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

[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0]

[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]

[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]

[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]

[7,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]

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

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

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

[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0]

[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]

[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]

[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]

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

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

[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]

[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]

[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]

[8,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]

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click to show experimental identities (only identities of compositions of up to three maps are shown)

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.

Properties

injective

Sage code

def mapping(la):
    if not la:
        return ParallelogramPolyomino([[1],[1]]).to_dyck_word()
    lower_path = [0]*la[0] + [1]*len(la)
    upper_path = []
    la = list(la) + [0]
    for i in range(len(la)-1):
        upper_path.extend([1]+[0]*(la[i]-la[i+1]))
    return ParallelogramPolyomino([lower_path, upper_path]).to_dyck_word()

Weight

Created

May 07, 2021 at 14:19 by Martin Rubey

Updated

May 07, 2021 at 14:19 by Martin Rubey