- FindStat

edit this statistic or download as text // json

Identifier

St001894: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 2

[-1,2] => 1

[-1,-2] => 3

[2,1] => 1

[2,-1] => 2

[-2,1] => 2

[-2,-1] => 2

[1,2,3] => 0

[1,2,-3] => 3

[1,-2,3] => 2

[1,-2,-3] => 5

[-1,2,3] => 1

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

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

[-1,-2,-3] => 6

[1,3,2] => 1

[1,3,-2] => 3

[1,-3,2] => 3

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

[-1,3,2] => 2

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

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

[-1,-3,-2] => 5

[2,1,3] => 1

[2,1,-3] => 4

[2,-1,3] => 2

[2,-1,-3] => 5

[-2,1,3] => 2

[-2,1,-3] => 5

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

[-2,-1,-3] => 5

[2,3,1] => 2

[2,3,-1] => 3

[2,-3,1] => 4

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

[-2,3,1] => 3

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

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

[-2,-3,-1] => 5

[3,1,2] => 2

[3,1,-2] => 4

[3,-1,2] => 3

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

[-3,1,2] => 3

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

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

[-3,-1,-2] => 5

[3,2,1] => 2

[3,2,-1] => 3

[3,-2,1] => 4

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

[-3,2,1] => 3

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

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

[-3,-2,-1] => 5

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

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

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

[1,2,-3,-4] => 7

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

[1,-2,3,-4] => 6

[1,-2,-3,4] => 5

[1,-2,-3,-4] => 9

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

[-1,2,3,-4] => 5

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

[-1,2,-3,-4] => 8

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

[-1,-2,3,-4] => 7

[-1,-2,-3,4] => 6

[-1,-2,-3,-4] => 10

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

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

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

[1,2,-4,-3] => 6

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

[1,-2,4,-3] => 6

[1,-2,-4,3] => 6

[1,-2,-4,-3] => 8

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

[-1,2,4,-3] => 5

[-1,2,-4,3] => 5

[-1,2,-4,-3] => 7

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

[-1,-2,4,-3] => 7

[-1,-2,-4,3] => 7

[-1,-2,-4,-3] => 9

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

[1,3,2,-4] => 5

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

[1,3,-2,-4] => 7

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

[1,-3,2,-4] => 7

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

[1,-3,-2,-4] => 8

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

[-1,3,2,-4] => 6

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

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.

References

[1] Petersen, T. K., Tenner, B. E. The depth of a permutation arXiv:1202.4765
[2] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups arXiv:1507.01180

Code

def statistic(sigma):
    """
    sage: sigma = SignedPermutation([2,-4,3,-1,7,-5,6])
    sage: statistic(sigma)
    12
    """
    s1 = sum(e - i for i, e in enumerate(sigma, 1) if e > i)
    s2 = sum(-e for e in sigma if e < 0)
    s3 = (oddness(sigma) - sigma.colors().count(-1)) / 2
    return s1 + s2 + s3

def oddness(sigma):
    """
    sage: sigma = SignedPermutation([2,-4,3,-1,7,-5,6])
    sage: oddness(sigma)
    1
    sage: sigma = SignedPermutation([2,-4,3,-6,7,-5,1])
    sage: oddness(sigma)
    1

    sage: sigma = SignedPermutation([-1,-2,-3,-4])
    sage: oddness(sigma)
    4
    """
    return sum(1 for b in decomposition(sigma)
               if is_odd(b.colors().count(-1)))

def decomposition(sigma):
    """
    sage: sigma = SignedPermutation([2,-4,3,-1,7,-5,6])
    sage: decomposition(sigma)
    [[2, -4, 3, -1], [3, -1, 2]]

    sage: sigma = SignedPermutation([2,-4,3,-6,7,-5,1])
    sage: decomposition(sigma)
    [[2, -4, 3, -6, 7, -5, 1]]
    """
    pi = sigma.permutation()
    summands = []
    j = 0
    for i in range(1, len(pi)):
        if max(pi[:i]) <= i:
            summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:i]])
            summands.append(summand)
            j = i
    summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:]])
    summands.append(summand)
    return summands

Created

Mar 31, 2023 at 20:59 by Martin Rubey

Updated

Mar 31, 2023 at 20:59 by Martin Rubey