Identifier
- St001078: Permutations ⟶ ℤ
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[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] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 3
[1,5,2,3,4] => 1
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 3
[1,5,4,2,3] => 3
[1,5,4,3,2] => 4
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 4
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[2,3,4,1,5] => 1
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,4,1] => 1
[2,4,1,3,5] => 3
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 1
[2,4,5,1,3] => 1
[2,4,5,3,1] => 2
[2,5,1,3,4] => 2
[2,5,1,4,3] => 3
[2,5,3,1,4] => 3
[2,5,3,4,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 3
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[3,1,4,2,5] => 3
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,4,2] => 3
[3,2,1,4,5] => 3
[3,2,1,5,4] => 4
[3,2,4,1,5] => 2
[3,2,4,5,1] => 1
[3,2,5,1,4] => 3
[3,2,5,4,1] => 2
[3,4,1,2,5] => 2
[3,4,1,5,2] => 1
[3,4,2,1,5] => 3
[3,4,2,5,1] => 2
[3,4,5,1,2] => 0
[3,4,5,2,1] => 1
[3,5,1,2,4] => 1
[3,5,1,4,2] => 2
[3,5,2,1,4] => 2
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Description
The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,...,n).
Let $\rho=(1,\dots,n)$ and $\sigma=(1,2)$. Then, for a permutation $\pi\in\mathfrak S_n$, this statistic is
$$\min\{k\mid \pi=\rho^{i_0}\sigma\rho^{i_1}\sigma\dots\rho^{i_{k-1}}\sigma\rho^{i_k}, 0\leq i_0,\dots,i_k < n\}.$$
Put differently, it is the minimal length of a factorization into cyclic shifts of the transposition $(1,2)$ (see St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).) of any of the permutations $\rho^k\pi$ for $0\leq k < n$.
Let $\rho=(1,\dots,n)$ and $\sigma=(1,2)$. Then, for a permutation $\pi\in\mathfrak S_n$, this statistic is
$$\min\{k\mid \pi=\rho^{i_0}\sigma\rho^{i_1}\sigma\dots\rho^{i_{k-1}}\sigma\rho^{i_k}, 0\leq i_0,\dots,i_k < n\}.$$
Put differently, it is the minimal length of a factorization into cyclic shifts of the transposition $(1,2)$ (see St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).) of any of the permutations $\rho^k\pi$ for $0\leq k < n$.
References
[1] Ryazanov, A. Generating $S_n$ with a fundamental transposition and a big cycle MathOverflow:290208
Code
@cached_function
def minimal_length(pi):
n = len(pi)
a = Permutation([2,1])
b = Permutation([i for i in range(2,n+1)] + [1])
powers = [Permutation([])]
for k in range(n-1):
powers.append(powers[-1]*b)
gens = [p.inverse()*a*p for p in powers]
G = PermutationGroup(gens)
pi = G(pi)
w = gap.Factorization(G._gap_(), pi._gap_())
return w.Length().sage()
def statistic(pi):
n = len(pi)
b = Permutation([i for i in range(2,n+1)] + [1])
powers = [Permutation([])]
for k in range(n-1):
powers.append(powers[-1]*b)
return min([minimal_length(p*pi) for p in powers])
Created
Jan 08, 2018 at 19:49 by Martin Rubey
Updated
Feb 09, 2021 at 15:28 by Martin Rubey
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