Inverse Permutation Research Articles

Normal Approximations for Descents and Inversions of Permutations of Multisets

Normal approximations for descents and inversions of permutations of the set {1,2,…,n} are well known. We consider the number of inversions of a permutation π(1),π(2),…,π(n) of a multiset with n elements, which is the number of pairs (i,j) with 1≤i π(j). The number of descents is the number of i in the range 1≤i π(i+1). We prove that, appropriately normalized, the distribution of both inversions and descents of a random permutation of the multiset approaches the normal distribution as n→∞, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than αn times in the multiset for a fixed α with 0<α<1. Both normal approximation theorems are proved using the size bias version of Stein’s method of auxiliary randomization and are accompanied by error bounds.

On edge-weighted recursive trees and inversions in random permutations

We introduce random recursive trees, where deterministically weights are attached to the edges according to the labeling of the trees. We will give a bijection between recursive trees and permutations, which relates the arising edge-weights in recursive trees with inversions of the corresponding permutations. Using this bijection we obtain exact and limiting distribution results for the number of permutations of size n, where exactly m elements have j inversions. Furthermore we analyze the distribution of the sum of labels of the elements, which have exactly j inversions, where we can identify Dickman's infinitely divisible distribution as the limit law. Moreover we give a distributional analysis of weighted depths and weighted distances in edge-weighted recursive trees.

Multidimensional Large‐Amplitude Motion: Revealing Concurrent Tunneling Pathways in Molecules with Several Internal Rotors

Light at the end of the tunnel: Knowledge of the feasible tunneling pathways of a molecule can shed light on the interaction mechanisms that govern its internal dynamics. A combination of rotational spectroscopy, permutation inversion group theory, and multidimensional tunneling formalism has shown that in the series (CH3)3XCl (X=Si, Ge, Sn), electrostatic repulsion is gradually reduced and the chemical bond gains importance.

Area of Catalan paths on a checkerboard

It is known that the area of all Catalan paths of length n is equal to 4 n − 2 n + 1 n , which coincides with the number of inversions of all 321-avoiding permutations of length n + 1 . In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented.

Propagation characteristics of [formula omitted] and Kloosterman sums

We study the inverse permutation σ : x ↦ x −1 on the field of order 2 n by means of their component functions f λ . We prove that the weights of derivatives of f λ can be expressed in terms of Kloosterman sums. We are then able to compute some indicators of the propagation characteristics of σ. We can claim that σ, which is considered as a good cryptographic mapping regarding several criteria, is moreover such that the functions f λ have good propagation properties with respect to these indicators. We further deduce several new formulas on Kloosterman sums, by using classical formulas which link any Boolean function with its derivatives.

On the number of inversions in bimodal permutations

Distributional properties of the random variable [ number of inversions] associated with random bimodal permutations are considered. Especially, the probability generating function and the moments (cumulants) are given and asymptotic properties are discussed.

Nuclear spin selection rules in chemical reactions by angular momentum algebra

The detailed selection rules for reactive collisions reported by Quack using molecular symmetry group are derived by using angular momentum algebra. Instead of the representations of the permutation–inversion group for both nuclear spin and rovibronic coordinate wavefunctions, those of the rotation group for nuclear spin wavefunction only are used. The method allows more straightforward derivation of Quack’s results and further extension of the calculation for separating elementary reactions and application to higher proton systems.

Anomalous splittings of torsional sublevels induced by the aldehyde inversion motion in the S1 state of acetaldehyde

The G6 group-theoretical high-barrier formalism developed previously for internally rotating and inverting CH3NHD is used to interpret the abnormal torsional splittings in the S1 state of acetaldehyde for levels 14(0-)15(0), 14(0-)15(1), and 14(0-)15(2), where 14(0-) denotes the upper inversion tunneling component of the aldehyde hydrogen and 15 denotes the methyl torsional vibration. This formalism, derived using an extended permutation-inversion group G6m, treats simultaneously methyl torsional tunneling, aldehyde-hydrogen inversion tunneling and overall rotation. Fits to the rotational states of the four pairs of inversion-torsion vibrational levels (14(0+)15(0A,E), 14(0-)15(0A,E)), (14(0+)15(1A,E), 14(0-)15(1A,E)), (14(0+)15(2A,E), 14(0-)15(2A,E)), and (14(0+)15(3A,E), 14(0-)15(3A,E)) are performed, giving root-mean-square deviations of 0.003, 0.004, 0.004, and 0.004 cm(-1), respectively, which are nearly equal to the experimental uncertainty of 0.003 cm(-1). For torsional levels lying near the top of the torsional barrier, this theoretical model, after including higher-order terms, provides satisfactory fits to the experimental data. The partially anomalous K-doublet structure of the S1 state, which deviates from that in a simple torsion-rotation molecule, is fitted using this formalism and is shown to arise from coupling of torsion and rotation motion with the aldehyde-hydrogen inversion.

Symmetry analysis of internal rotation

Research papers and textbooks addressing the problem of internal rotation in a molecule explain symmetry properties of the torsional potential by local geometrical symmetries of the molecule. It is shown here that symmetry properties of a torsional potential derive from permutation inversion symmetry and a peculiar nature of torsional dynamics but have no relation to actual geometrical symmetries. To confirm the validity of our symmetry analysis a minimum energy torsional potential curve has been determined ab initio for acetaldehyde, resulting in exact 2π/3 periodicity that no previous ab initio calculations achieved.

Application of Permutation–Inversion Group Theory to the Interpretation of the Microwave Absorption Spectrum of Dimethyl Methylphosphonate

The G36 permutation–inversion group theoretical tunneling–rotational formalism originally developed for the methanol dimer has been modified (for the subgroup G18) and extended (to the larger group G54) for application to dimethyl methylphosphonate, CH3P(O)(OCH3)2, which has three large-amplitude methyl top internal rotation motions and one large-amplitude methoxy interchange motion. Energy levels of this chiral molecule are conveniently labeled by symmetry species corresponding to a mixed set of irreducible and reducible representations of G18 denoted by A1, A2, E, E1sep, E2sep, and Gsep. The separably degenerate species (with subscript sep) consist of pairs of irreducible representations of G18 whose energies are degenerate for Hamiltonians invariant to time reversal. All characters of these separably degenerate representations are real. Comparison of the group-theoretically derived splitting patterns with Fourier transform microwave and ab initio results from the preceding paper permit drawing a semiquantitative energy level diagram showing how a given Ka=0 level splits into A1⊕A2⊕2E⊕E1sep⊕E2sep⊕2Gsep components when the large-amplitude motions are turned on in the following order: (i) low-barrier methyl top internal rotation, (ii) medium-barrier methyl top internal rotation, (iii) top–top interaction, and (iv) methoxy interchange motion. (Internal rotation of the high-barrier methyl top is ignored.) Spectral splitting patterns observed for Ka=1–1 transitions are also quite regular, being either the same as, or mirror images of, the Ka=0–0 patterns. Theoretical work on Ka>0 splitting patterns is in progress.

Shared Processes in Spatial Rotation and Musical Permutation

An experiment was conducted in which subjects performed a three-dimensional spatial rotation test (24 trials) and a new test involving judgments of musical permutations (64 trials). Two types of musical permutations were used, including retrograde and inverse. In a retrograde permutation, the criterion melody was played backward in the test melody, and in an inverse permutation, an ascending or descending interval in the criterion melody became an opposite in the test melody. Subjects included 32 male and 64 female undergraduates at the University of Toronto. Regression analysis clearly showed that it was easiest to compare short retrograde permutations and that accuracy at discerning retrograde permutations predicted accuracy at judging spatial rotations. The implication is that a higher order ability to discriminate contour underlies both kinds of judgments.

Structure and rovibrational analysis of the [O2(1Δg)v=0]2←[O2(3Σg−)v=0]2 transition of the O2 dimer

The rotationally resolved absorption spectrum of the O2 dimer involving the [O2(1Δg)v=0]2←[O2(3Σg−)v=0]2 transition has been recorded near 632.6 nm by continuous wave Cavity Ring Down Spectroscopy in a supersonic slit jet expansion of pure O2. A quadratic dependence of the absorption in the jet versus the stagnation pressure is observed. A rotational temperature of 12 K is derived from the (O2)2 rotational analysis. The high spectral resolution of the CW-CRDS measurements limited by the residual Doppler broadening in the jet and the low rotational temperature allow the first rotational analysis in this open-shell complex. The same spectrum was also recorded by Intracavity Laser Absorption Spectroscopy and the comparison of the performances of the two methods is discussed. Among more than 600 lines measured between 15 800 and 15 860 cm−1 from the CW-CRDS spectrum, 40 were assigned to the RP0, RQ0, and RR0 branches of two subbands associated with B1−←A1+ and A1+←B1− transitions between the ground and excited rovibrational levels, labeled following the G16 permutation inversion representation. Forty five lines were assigned to PP2, PQ2, and PR2 branches of two subbands associated with B1−←A1+ and A1+←B1− transitions. The subbands centered at 15 808.401(49) [A1+←B1−] and 15 813.134(37) cm−1 [B1−←A1+] for those arising from K=0, and at 15 812.656(20) [A1+←B1−] and 15 818.277(35) [B1−←A1+] when arising from K=2, are analyzed considering (O2)2 as a slightly asymmetric prolate top. The rotational analysis of the two K=0 subbands leads to very close values of the effective rotational constant, Bp=(B+C)/2, for both A1+ and B1− levels: 0.095 cm−1 for the [O2(3Σg−)v=0]2 lower states and 0.063 cm−1 for the [O2(1Δg)v=0]2 excited states, in close agreement with theoretical values. The H geometry is confirmed as the most stable for the ground electronic singlet state. A distance between the two monomers of 6.1 a0 and 7.5 a0 is derived for the ground and excited singlet states. Similar results are obtained from the two K=2 subbands. A vibrational assignment is given for the two rotationally analyzed subbands (K=0) and proposed for the main features of the whole band.

Some combinatorial interpretations ofq-analogs of Schröder numbers

We introduce two definitions of Schroder numberq-analogs and show some combinatorial interpretations of theseq-numbers. We use the following combinatorial objects for these interpretations: Schroder paths, 1-colored parallelogram polyominoes and permutations with forbidden subsequences (4231, 4132). We enumerate these objects according to various parameters by means of a recentq-counting technique. We prove that the firstq-Schroder number enumerates of Schroder paths with respect to area and the number of permutation inversions, while the second one counts the 1-colored parallelogram polyominoes according to their width and area. Finally, we illustrate some relations among the parameters characterizing the combinatorial objects.

Inversions in k-sorted permutations

When a list of size n is nearly sorted, a straight insertion sort algorithm is highly efficient since only a number of comparisons equal to the number of inversions in the original list, plus at most n − 1, is required. We use a definition of nearly sorted, k-sorted, as given in Berman (1997) and determine the maximum number of inversions in k-sorted permutations of size n. This number is approximately 0.6 kn.

A Combinatorial Approach to the Fusion Process for the Symmetric Group

We give a detailed account of Cherednik's fusion process for the symmetric group using as a key tool the combinatorics ofcompatible orderson the set of inversions of permutations.

Public key encryption and digital signatures basedon permutation polynomials

Permutation polynomials over Zn form the basis of the RSA and Dickson public key schemes. Previously, however, only those permutation polynomials whose inverse permutation polynomial was easy to evaluate have been used in cryptography. The authors propose a way to avoid this restriction in public key cryptography by implementing secret key decryption and signature generation by computation of the gcd of two polynomials. This allows the implementation of new classes of public key scheme.

Shellsort with three increments

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is $O(n^{23/15})$. The proof involves interesting properties of the inversions in random permutations that have been $h$-sorted and $g$-sorted.

Symmetry Analysis of Molecules Consisting of Two Coaxial Rotors Using the Extended Permutation–Inversion Groups

A general method for constructing permutation-inversion groups and their extensions to molecules containing two coaxial rotors is presented. Symmetry species of the dipole moment, tensor of polarizability, and total angular momentum operator are determined. General infrared and Raman selection rules, symmetry classifications of rotorsional, rotational, and torsional wavefunctions are derived. Theory is illustrated for the H3D+2complex.

A Preliminary Study of the Proton Rearrangement Energy Levels and Spectrum of CH+5

This theoretical paper is concerned with the proton rearrangement energy levels and spectrum of the CH+5molecular ion, and it is based on theab initioresults of P. R. Schreiner, S-J. Kim, H. F. Schaefer, and P. v. R. Schleyer [J. Chem. Phys.99,3716–3720 (1993)]. Theab initiowork predicts that the molecule should be considered as an H2molecule bound with a dissociation energy of about 15000 cm−1at the apex of a pyramidal CH+3group. At equilibrium the H2axis is nearly perpendicular to theC3axis of the CH+3group, eclipsing a CH bond. The internal rotation of the H2about theC3axis has a barrier height of 30 cm−1. There is also an internal “flip” motion through aC2vstructure, with a barrier of 300 cm−1, that exchanges a CH+3and an H2proton in the molecule and that makes all 120 symmetrically equivalent minima accessible. Using theab initioequilibrium structure and torsional barrier, the rotation–torsion energy levels and spectrum are calculated. The tilt of theC3axis of the CH+3group away from the internal rotation axis of the H2has a very significant effect on the energy levels. The tunneling resulting from the flip motion will produce splittings in the rotation–torsion energy levels and a spectrum that will have characteristic relative intensities because of the nuclear spin statistical weights. These weights are calculated using the complete nuclear permutation inversion groupG240= S5× {E,E*}, all elements of which are feasible. There are many levels with zero nuclear spin statistical weight, and this will make the spectrum simpler than would otherwise be the case.

Group theoretical study of the radical cation of methane: The effect of tunneling motions on the hyperfine interaction

The electron spin resonance (ESR) spectrum of CH+4 indicates that some large amplitude tunneling motions among Jahn–Teller distorted structures make the four protons equivalent. A group theoretical study using the permutation–inversion (PI) group is performed to analyze the hyperfine interaction of the nonrigid CH+4. It is shown that three patterns of the interaction are possible depending upon the type of tunneling motions. Only one of the three patterns is consistent with the experimental spectrum, which is presented in the accompanying paper [J. Chem. Phys. 103, 3377 (1995)].