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Identifier

St000520: Permutations ⟶ ℤ

Values

[1,2] => 3

[2,1] => 3

[1,2,3] => 4

[1,3,2] => 5

[2,1,3] => 5

[2,3,1] => 5

[3,1,2] => 5

[3,2,1] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 6

[1,2,3,5,4] => 9

[1,2,4,3,5] => 11

[1,2,4,5,3] => 11

[1,2,5,3,4] => 11

[1,2,5,4,3] => 10

[1,3,2,4,5] => 11

[1,3,2,5,4] => 11

[1,3,4,2,5] => 13

[1,3,4,5,2] => 11

[1,3,5,2,4] => 15

[1,3,5,4,2] => 13

[1,4,2,3,5] => 13

[1,4,2,5,3] => 15

[1,4,3,2,5] => 12

[1,4,3,5,2] => 14

[1,4,5,2,3] => 12

[1,4,5,3,2] => 12

[1,5,2,3,4] => 11

[1,5,2,4,3] => 13

[1,5,3,2,4] => 14

[1,5,3,4,2] => 14

[1,5,4,2,3] => 12

[1,5,4,3,2] => 9

[2,1,3,4,5] => 9

[2,1,3,5,4] => 11

[2,1,4,3,5] => 11

[2,1,4,5,3] => 12

[2,1,5,3,4] => 12

[2,1,5,4,3] => 10

[2,3,1,4,5] => 11

[2,3,1,5,4] => 12

[2,3,4,1,5] => 11

[2,3,4,5,1] => 9

[2,3,5,1,4] => 14

[2,3,5,4,1] => 12

[2,4,1,3,5] => 15

[2,4,1,5,3] => 15

[2,4,3,1,5] => 14

[2,4,3,5,1] => 14

[2,4,5,1,3] => 14

[2,4,5,3,1] => 13

[2,5,1,3,4] => 14

[2,5,1,4,3] => 14

[2,5,3,1,4] => 16

[2,5,3,4,1] => 14

[2,5,4,1,3] => 14

[2,5,4,3,1] => 11

[3,1,2,4,5] => 11

[3,1,2,5,4] => 12

[3,1,4,2,5] => 15

[3,1,4,5,2] => 14

[3,1,5,2,4] => 15

[3,1,5,4,2] => 14

[3,2,1,4,5] => 10

[3,2,1,5,4] => 10

[3,2,4,1,5] => 13

[3,2,4,5,1] => 12

[3,2,5,1,4] => 14

[3,2,5,4,1] => 12

[3,4,1,2,5] => 12

[3,4,1,5,2] => 14

[3,4,2,1,5] => 12

[3,4,2,5,1] => 13

[3,4,5,1,2] => 10

[3,4,5,2,1] => 10

[3,5,1,2,4] => 14

[3,5,1,4,2] => 15

[3,5,2,1,4] => 14

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Description

The number of patterns in a permutation.
In other words, this is the number of subsequences which are not order-isomorphic.

Code

def statistic(pi):
    res = []
    for w in Subwords(pi):
        dom = sorted(w)
        res += [Permutation([dom.index(e)+1 for e in w])]
    return len(set(res))

#alternative faster code
@cached_function
def reduce_word(w):
    dom = { j:i for i,j in enumerate(sorted(w)) }
    return tuple(dom[j] for j in w)

def statistic_alternative(pi):
    res = set()
    for w in Subwords(pi):
        res.add(reduce_word(tuple(w)))
    return len(res)

Created

Jun 01, 2016 at 14:50 by Martin Rubey

Updated

Oct 17, 2017 at 11:00 by Christian Stump