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Identifier
Values
=>
[1]=>1 [1,2]=>1 [2,1]=>1 [1,2,3]=>1 [1,3,2]=>1 [2,1,3]=>1 [2,3,1]=>2 [3,1,2]=>2 [3,2,1]=>4 [1,2,3,4]=>1 [1,2,4,3]=>1 [1,3,2,4]=>1 [1,3,4,2]=>2 [1,4,2,3]=>2 [1,4,3,2]=>4 [2,1,3,4]=>1 [2,1,4,3]=>2 [2,3,1,4]=>2 [2,3,4,1]=>6 [2,4,1,3]=>6 [2,4,3,1]=>16 [3,1,2,4]=>2 [3,1,4,2]=>6 [3,2,1,4]=>4 [3,2,4,1]=>16 [3,4,1,2]=>20 [3,4,2,1]=>52 [4,1,2,3]=>6 [4,1,3,2]=>16 [4,2,1,3]=>16 [4,2,3,1]=>64 [4,3,1,2]=>52 [4,3,2,1]=>168 [1,2,3,4,5]=>1 [1,2,3,5,4]=>1 [1,2,4,3,5]=>1 [1,2,4,5,3]=>2 [1,2,5,3,4]=>2 [1,2,5,4,3]=>4 [1,3,2,4,5]=>1 [1,3,2,5,4]=>2 [1,3,4,2,5]=>2 [1,3,4,5,2]=>6 [1,3,5,2,4]=>6 [1,3,5,4,2]=>16 [1,4,2,3,5]=>2 [1,4,2,5,3]=>6 [1,4,3,2,5]=>4 [1,4,3,5,2]=>16 [1,4,5,2,3]=>20 [1,4,5,3,2]=>52 [1,5,2,3,4]=>6 [1,5,2,4,3]=>16 [1,5,3,2,4]=>16 [1,5,3,4,2]=>64 [1,5,4,2,3]=>52 [1,5,4,3,2]=>168 [2,1,3,4,5]=>1 [2,1,3,5,4]=>2 [2,1,4,3,5]=>2 [2,1,4,5,3]=>6 [2,1,5,3,4]=>6 [2,1,5,4,3]=>16 [2,3,1,4,5]=>2 [2,3,1,5,4]=>6 [2,3,4,1,5]=>6 [2,3,4,5,1]=>24 [2,3,5,1,4]=>24 [2,3,5,4,1]=>80 [2,4,1,3,5]=>6 [2,4,1,5,3]=>24 [2,4,3,1,5]=>16 [2,4,3,5,1]=>80 [2,4,5,1,3]=>100 [2,4,5,3,1]=>312 [2,5,1,3,4]=>24 [2,5,1,4,3]=>80 [2,5,3,1,4]=>80 [2,5,3,4,1]=>384 [2,5,4,1,3]=>312 [2,5,4,3,1]=>1176 [3,1,2,4,5]=>2 [3,1,2,5,4]=>6 [3,1,4,2,5]=>6 [3,1,4,5,2]=>24 [3,1,5,2,4]=>24 [3,1,5,4,2]=>80 [3,2,1,4,5]=>4 [3,2,1,5,4]=>16 [3,2,4,1,5]=>16 [3,2,4,5,1]=>80 [3,2,5,1,4]=>80 [3,2,5,4,1]=>320 [3,4,1,2,5]=>20 [3,4,1,5,2]=>100 [3,4,2,1,5]=>52 [3,4,2,5,1]=>312 [3,4,5,1,2]=>464 [3,4,5,2,1]=>1408 [3,5,1,2,4]=>100 [3,5,1,4,2]=>424 [3,5,2,1,4]=>312 [3,5,2,4,1]=>1752 [3,5,4,1,2]=>1680 [3,5,4,2,1]=>6016 [4,1,2,3,5]=>6 [4,1,2,5,3]=>24 [4,1,3,2,5]=>16 [4,1,3,5,2]=>80 [4,1,5,2,3]=>100 [4,1,5,3,2]=>312 [4,2,1,3,5]=>16 [4,2,1,5,3]=>80 [4,2,3,1,5]=>64 [4,2,3,5,1]=>384 [4,2,5,1,3]=>424 [4,2,5,3,1]=>1752 [4,3,1,2,5]=>52 [4,3,1,5,2]=>312 [4,3,2,1,5]=>168 [4,3,2,5,1]=>1176 [4,3,5,1,2]=>1680 [4,3,5,2,1]=>6016 [4,5,1,2,3]=>464 [4,5,1,3,2]=>1680 [4,5,2,1,3]=>1680 [4,5,2,3,1]=>9216 [4,5,3,1,2]=>6720 [4,5,3,2,1]=>27968 [5,1,2,3,4]=>24 [5,1,2,4,3]=>80 [5,1,3,2,4]=>80 [5,1,3,4,2]=>384 [5,1,4,2,3]=>312 [5,1,4,3,2]=>1176 [5,2,1,3,4]=>80 [5,2,1,4,3]=>320 [5,2,3,1,4]=>384 [5,2,3,4,1]=>2176 [5,2,4,1,3]=>1752 [5,2,4,3,1]=>8032 [5,3,1,2,4]=>312 [5,3,1,4,2]=>1752 [5,3,2,1,4]=>1176 [5,3,2,4,1]=>8032 [5,3,4,1,2]=>9216 [5,3,4,2,1]=>37312 [5,4,1,2,3]=>1408 [5,4,1,3,2]=>6016 [5,4,2,1,3]=>6016 [5,4,2,3,1]=>37312 [5,4,3,1,2]=>27968 [5,4,3,2,1]=>130560
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Description
The number of Bruhat factorizations of a permutation.
This is the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that the number of inversions of $t_1 \cdots t_i$ equals $i$ for all $1 \leq i \leq \ell$.
Code
@cached_function
def bruhat_poset(n):
    return Permutations(n).bruhat_poset(facade=True)

def statistic(pi):
    P = bruhat_poset(len(pi))
    I = P.subposet(P.principal_order_ideal(pi))
    return len(I.maximal_chains())
Created
Aug 28, 2017 at 11:08 by Christian Stump
Updated
Jan 13, 2018 at 12:45 by Martin Rubey