- FindStat

edit this map or download as text // json

Identifier

Mp00095: Integer partitions —to binary word⟶ Binary words

Images

[] =>

[1] => 10

[2] => 100

[1,1] => 110

[3] => 1000

[2,1] => 1010

[1,1,1] => 1110

[4] => 10000

[3,1] => 10010

[2,2] => 1100

[2,1,1] => 10110

[1,1,1,1] => 11110

[5] => 100000

[4,1] => 100010

[3,2] => 10100

[3,1,1] => 100110

[2,2,1] => 11010

[2,1,1,1] => 101110

[1,1,1,1,1] => 111110

[6] => 1000000

[5,1] => 1000010

[4,2] => 100100

[4,1,1] => 1000110

[3,3] => 11000

[3,2,1] => 101010

[3,1,1,1] => 1001110

[2,2,2] => 11100

[2,2,1,1] => 110110

[2,1,1,1,1] => 1011110

[1,1,1,1,1,1] => 1111110

[7] => 10000000

[6,1] => 10000010

[5,2] => 1000100

[5,1,1] => 10000110

[4,3] => 101000

[4,2,1] => 1001010

[4,1,1,1] => 10001110

[3,3,1] => 110010

[3,2,2] => 101100

[3,2,1,1] => 1010110

[3,1,1,1,1] => 10011110

[2,2,2,1] => 111010

[2,2,1,1,1] => 1101110

[2,1,1,1,1,1] => 10111110

[1,1,1,1,1,1,1] => 11111110

[8] => 100000000

[7,1] => 100000010

[6,2] => 10000100

[6,1,1] => 100000110

[5,3] => 1001000

[5,2,1] => 10001010

[5,1,1,1] => 100001110

[4,4] => 110000

[4,3,1] => 1010010

[4,2,2] => 1001100

[4,2,1,1] => 10010110

[4,1,1,1,1] => 100011110

[3,3,2] => 110100

[3,3,1,1] => 1100110

[3,2,2,1] => 1011010

[3,2,1,1,1] => 10101110

[3,1,1,1,1,1] => 100111110

[2,2,2,2] => 111100

[2,2,2,1,1] => 1110110

[2,2,1,1,1,1] => 11011110

[2,1,1,1,1,1,1] => 101111110

[1,1,1,1,1,1,1,1] => 111111110

[9] => 1000000000

[8,1] => 1000000010

[7,2] => 100000100

[7,1,1] => 1000000110

[6,3] => 10001000

[6,2,1] => 100001010

[6,1,1,1] => 1000001110

[5,4] => 1010000

[5,3,1] => 10010010

[5,2,2] => 10001100

[5,2,1,1] => 100010110

[5,1,1,1,1] => 1000011110

[4,4,1] => 1100010

[4,3,2] => 1010100

[4,3,1,1] => 10100110

[4,2,2,1] => 10011010

[4,2,1,1,1] => 100101110

[4,1,1,1,1,1] => 1000111110

[3,3,3] => 111000

[3,3,2,1] => 1101010

[3,3,1,1,1] => 11001110

[3,2,2,2] => 1011100

[3,2,2,1,1] => 10110110

[3,2,1,1,1,1] => 101011110

[3,1,1,1,1,1,1] => 1001111110

[2,2,2,2,1] => 1111010

[2,2,2,1,1,1] => 11101110

[2,2,1,1,1,1,1] => 110111110

[2,1,1,1,1,1,1,1] => 1011111110

[1,1,1,1,1,1,1,1,1] => 1111111110

[10] => 10000000000

[9,1] => 10000000010

[8,2] => 1000000100

[8,1,1] => 10000000110

>>> Load all 1578 entries. <<<

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

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.

Properties

injective

Sage code

def mapping(p):
    if not p:
        return Words([0,1])([])
    res = [1]    
    for (part,nextpart) in zip(p, p[1:]):
        res += [0]*(part-nextpart) + [1]
        
    return Words([0,1])(res + [0]*p[-1])

Weight

Created

Jan 29, 2020 at 13:25 by FindStatCrew

Updated

Jan 29, 2020 at 13:25 by Martin Rubey