Permutations and Negative Beta-Shifts

Abstract

Elizalde (2011) characterized which permutations can be obtained by ordering consecutive elements in the trajectories of (positive) beta-transformations and beta-shifts. We prove similar results for negative bases beta. 1 n log #{π ∈ S n : B − (π) < β} = log β (which is the entropy of the (−β)-transformation) for β > 1, but we do not know whether c n = #{π ∈ S n : B − (π) = 1} grows polynomially; we have c 2 = 2, c 3 = 5, c 4 = 12, c 5 = 19, c 6 = 34, c 7 = 57, c 8 = 82, c 9 = 115, . . .Recall that #{π ∈ S n : B + (π) = 1} = n. It would also be interesting to give more precise asymptotics for the number of permutations with B − (π) < N or B − (π) ≤ N for some integer N ≥ 2. The following theorem characterizes B − (π) < N .Theorem 3. Let π ∈ S n , n ≥ 2. The minimal number of distinct elements of a sequence w satisfying Pat(w, Σ, n) = π, w.r.t. the alternating lexicographical order, is N − (π) = ⌊B − (π)⌋ + 1 = max 1≤j<n z j + 1 + ǫ, with ǫ = 1 if π is collapsed or a = max 1≤j<n z j 0, ǫ = 0 otherwise. We have N −

Comparison of methods for intravoxel incoherent motion parameter estimation in the brain from flow‐compensated and non‐flow‐compensated diffusion‐encoded data

“…Difference in estimated IVIM parameter values and their repeatability among estimation methods were tested with the Friedman test using the exact method to calculate the test statistic with the implementation in the Python package permutation‐stats 31 . When the Friedman test was statistically significant ( p < 0.05), pairwise differences between methods were tested with the signed‐rank test.…”

Characterizations and enumerations of patterns of signed shifts

“…Therefore, w [n,∞) must begin with d, and by canceling equal prefixes and repeating this process, we determine that the only option would be w [n,∞) = d ∞ , which does not satisfy the inequalities (7). We conclude that w [x,∞) < σ w [l,∞) , and so S(x, l) holds.…”

“…The permutation satisfying (i), (ii), (iii) is π = 5173264. Next, Equation (18) with z ′ [1,7] = 1021021 yields F = (0, 2, 5, 7). Finally, we add additional indices to F , with multiplicty, by including the index min{π m , π m+3 } for each m ∈ M. In this case, we include the index min{π 2 , π 5 } = 1, and two copies of the index min{π 3 , π 6 } = 6.…”

“…A characterization of allowed patterns when f (x) = {βx} for a real number β > 1 was given in [9]. The case of negative β has been addressed recently in [7] and [11].…”

“…After we posted an earlier preprint including this result on arxiv.org, we were informed by Charlier and Steiner that they independently obtained Theorem 5.3 and Corollary 5.5. Their work was subsequently posted in[7].…”