Identifier
- St001685: Permutations ⟶ ℤ
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 3
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,5,2,3,4] => 1
[1,5,2,4,3] => 2
[1,5,3,2,4] => 1
[1,5,3,4,2] => 1
[1,5,4,2,3] => 1
[1,5,4,3,2] => 1
[2,1,3,4,5] => 0
[2,1,3,5,4] => 3
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 2
[2,3,1,4,5] => 0
[2,3,1,5,4] => 3
[2,3,4,1,5] => 0
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 1
[2,4,1,5,3] => 2
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 1
[2,4,5,3,1] => 1
[2,5,1,3,4] => 1
[2,5,1,4,3] => 2
[2,5,3,1,4] => 1
[2,5,3,4,1] => 1
[2,5,4,1,3] => 1
[2,5,4,3,1] => 1
[3,1,2,4,5] => 0
[3,1,2,5,4] => 3
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 0
[3,2,1,5,4] => 3
[3,2,4,1,5] => 0
[3,2,4,5,1] => 0
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 0
[3,4,1,5,2] => 1
[3,4,2,1,5] => 0
[3,4,2,5,1] => 0
[3,4,5,1,2] => 0
[3,4,5,2,1] => 0
[3,5,1,2,4] => 1
[3,5,1,4,2] => 2
>>> Load all 1200 entries. <<<
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Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
References
[1] Thamrongpairoj, S., Remmel, J. B. Positional Marked Patterns in Permutations arXiv:2102.03867
Code
def statistic(pi, tau=[1,3,2], mark=0):
"""
The number of positions i such that tau occurs in pi and the mark
is at position i.
EXAMPLES::
sage: pi = Permutation([2,6,4,8,1,5,7,3])
sage: tau = Permutation([1,4,3,2])
sage: statistic(pi, tau, 1)
2
Table before section 3.2::
sage: q = QQ["q"].gen()
sage: [sum(q^statistic(pi, [1,2,3], 0) for pi in Permutations(n)) for n in range(1,7)]
[1,
2,
q + 5,
2*q^2 + 8*q + 14,
6*q^3 + 25*q^2 + 47*q + 42,
24*q^4 + 104*q^3 + 216*q^2 + 244*q + 132]
"""
return len(set([pos[mark] for pos in pi.pattern_positions(tau)]))
Created
Feb 20, 2021 at 18:00 by Martin Rubey
Updated
Feb 20, 2021 at 18:00 by Martin Rubey
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