- FindStat

Identifier

St001437: Binary words ⟶ ℤ

Values

0 => 1

1 => 1

00 => 2

01 => 1

10 => 2

11 => 2

000 => 3

001 => 1

010 => 2

011 => 1

100 => 3

101 => 2

110 => 3

111 => 3

0000 => 4

0001 => 1

0010 => 2

0011 => 1

0100 => 3

0101 => 2

0110 => 2

0111 => 1

1000 => 4

1001 => 3

1010 => 4

1011 => 2

1100 => 4

1101 => 3

1110 => 4

1111 => 4

00000 => 5

00001 => 1

00010 => 2

00011 => 1

00100 => 3

00101 => 1

00110 => 2

00111 => 1

01000 => 4

01001 => 2

01010 => 3

01011 => 1

01100 => 3

01101 => 2

01110 => 2

01111 => 1

10000 => 5

10001 => 4

10010 => 4

10011 => 3

10100 => 5

10101 => 3

10110 => 4

10111 => 2

11000 => 5

11001 => 4

11010 => 5

11011 => 3

11100 => 5

11101 => 4

11110 => 5

11111 => 5

000000 => 6

000001 => 1

000010 => 2

000011 => 1

000100 => 3

000101 => 1

000110 => 2

000111 => 1

001000 => 4

001001 => 2

001010 => 2

001011 => 1

001100 => 3

001101 => 1

001110 => 2

001111 => 1

010000 => 5

010001 => 3

010010 => 4

010011 => 2

010100 => 4

010101 => 3

010110 => 2

010111 => 1

011000 => 4

011001 => 3

011010 => 3

011011 => 2

011100 => 3

011101 => 2

011110 => 2

011111 => 1

100000 => 6

100001 => 5

100010 => 5

100011 => 4

100100 => 6

100101 => 4

100110 => 4

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Description

The flex of a binary word.
This is the product of the lex statistic (St001436The index of a given binary word in the lex-order among all its cyclic shifts., augmented by 1) and its frequency (St000627The exponent of a binary word.), see [1, §8].

References

[1] Ahlbach, C., Swanson, J. P. Refined cyclic sieving on words for the major index statistic MathSciNet:3836732

Code

def lex(w):
    orbit = [ tuple(w[i:] + w[:i]) for i in range(len(w)) ]
    return sorted(set(orbit)).index(tuple(w))+1

def freq(w):
    w = list(w)
    n = len(w)
    for i in sorted(Integer(n).divisors()):
        if w == w[:i]*(n//i):
            return n // i

def statistic(w):
    return lex(w)*freq(w)

Created

Jul 02, 2019 at 11:52 by Christian Stump

Updated

Jul 04, 2019 at 22:29 by Martin Rubey