Identifier
- St000958: Permutations ⟶ ℤ
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 $1 \leq i \leq \ell$.
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
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