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Identifier

Mp00316: Binary words —inverse Foata bijection⟶ Binary words

Images

0 => 0

1 => 1

00 => 00

01 => 01

10 => 10

11 => 11

000 => 000

001 => 001

010 => 100

011 => 011

100 => 010

101 => 101

110 => 110

111 => 111

0000 => 0000

0001 => 0001

0010 => 1000

0011 => 0011

0100 => 0100

0101 => 1001

0110 => 1100

0111 => 0111

1000 => 0010

1001 => 0101

1010 => 0110

1011 => 1011

1100 => 1010

1101 => 1101

1110 => 1110

1111 => 1111

00000 => 00000

00001 => 00001

00010 => 10000

00011 => 00011

00100 => 01000

00101 => 10001

00110 => 11000

00111 => 00111

01000 => 00100

01001 => 01001

01010 => 01100

01011 => 10011

01100 => 10100

01101 => 11001

01110 => 11100

01111 => 01111

10000 => 00010

10001 => 00101

10010 => 00110

10011 => 01011

10100 => 10010

10101 => 01101

10110 => 01110

10111 => 10111

11000 => 01010

11001 => 10101

11010 => 10110

11011 => 11011

11100 => 11010

11101 => 11101

11110 => 11110

11111 => 11111

000000 => 000000

000001 => 000001

000010 => 100000

000011 => 000011

000100 => 010000

000101 => 100001

000110 => 110000

000111 => 000111

001000 => 001000

001001 => 010001

001010 => 011000

001011 => 100011

001100 => 101000

001101 => 110001

001110 => 111000

001111 => 001111

010000 => 000100

010001 => 001001

010010 => 001100

010011 => 010011

010100 => 100100

010101 => 011001

010110 => 011100

010111 => 100111

011000 => 010100

011001 => 101001

011010 => 101100

011011 => 110011

011100 => 110100

011101 => 111001

011110 => 111100

011111 => 011111

100000 => 000010

100001 => 000101

100010 => 000110

100011 => 001011

100100 => 100010

100101 => 001101

100110 => 001110

>>> Load all 1074 entries. <<<

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

Description

The inverse of Foata's bijection.
See Mp00096Foata bijection.

References

[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852

Properties

bijective, graded

Sage code

def mapping(self):
    """
    sage: all(mapping(pi.foata_bijection()) == pi for n in range(9) for pi in Words([0,1], n))
    True
    """
    self = Words([0,1])(self)
    s = self.standard_permutation()
    ordered_alphabet = sorted(self.letters(),
                              key=self.parent().sortkey_letters)
    eval_dict = self.evaluation_dict()
    weight = [eval_dict[a] for a in ordered_alphabet]
    return Words([0,1])((s.foata_bijection_inverse()).destandardize(weight, ordered_alphabet=ordered_alphabet))

Weight

Created

Sep 12, 2023 at 11:34 by Martin Rubey

Updated

Sep 12, 2023 at 11:34 by Martin Rubey