Brant Jones

The enumeration of fully commutative affine permutations

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci-Del Lungo-Pergola-Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.

High-resolution Rydberg tagging time-of-flight measurements of atomic photofragments by single-photon vacuum ultraviolet laser excitation

By coupling a comprehensive tunable vacuum ultraviolet (VUV) laser system to a velocity-mapped ion imaging apparatus, we show that high-resolution high-n Rydberg tagging time-of-flight (TOF) measurements of nascent atomic photofragments formed by laser photodissociation can be made using single-photon VUV laser photoexcitation. To illustrate this single-photon Rydberg tagging TOF method, we present here the results of the VUV laser high-n Rydberg tagging TOF measurements of O((3)P(2)) and S((3)P(2)) formed in the photodissociation of SO(2) and CS(2) at 193.3 and 202.3 nm, respectively. These results are compared to those obtained by employing the VUV laser photoionization time-sliced velocity-mapped ion imaging technique. The fact that the kinetic energy resolutions achieved in the VUV laser high-n Rydberg tagging TOF measurements of O and S atoms are found to be higher than those observed in the VUV laser photoionization, time-sliced velocity-mapped ion imaging studies show that the single-photon VUV laser high-n Rydberg tagging TOF method is useful and complementary to state-of-the-art time-sliced velocity-mapped ion imaging measurements of heavier atomic photofragments, such as O and S atoms. Furthermore, the general agreement observed between the VUV laser high-n Rydberg tagging TOF and velocity-mapped ion imaging experiments supports the conclusion that the lifetimes of the tagged Rydberg states of O and S atoms are sufficiently long to allow the reliable determination of state-resolved UV photodissociation cross sections of SO(2) and CS(2) by using the VUV laser high-n Rydberg tagging TOF method.

Affine structures and a tableau model for E_6 crystals

Abstract We provide the unique affine crystal structure for type E 6 ( 1 ) Kirillov–Reshetikhin crystals corresponding to the multiples of fundamental weights s Λ 1 , s Λ 2 , and s Λ 6 for all s ⩾ 1 (in Bourbakis labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type E and use the order three automorphism of the affine E 6 ( 1 ) Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type E 7 ( 1 ) Kirillov–Reshetikhin crystals corresponding to the adjoint node.

A vacuum ultraviolet laser photoionization and pulsed field ionization study of nascent S(P2,1,03) and S(D21) formed in the 193.3nm photodissociation of CS2

The photoionization efficiency (PIE) and pulsed field ionization-photoion (PFI-PI) spectra for sulfur atoms S(3P2,1,0) and S(1D2) resulting from the 193.3 nm photodissociation of CS2 have been measured using tunable vacuum ultraviolet (vuv) laser radiation in the frequency range of 82 750-83 570 cm(-1). The PIE spectrum of S(3P2,1,0) near their ionization threshold exhibits steplike structures. On the basis of the velocity-mapped ion-imaging measurements, four strong autoionizing peaks observed in the PIE measurement in this frequency range have been identified to originate from vuv excitation of S(1D2). The PFI-PI measurement reveals over 120 previously unidentified new Rydberg lines. They have been assigned as Rydberg states [3p3(4S composite function nd3 D composite function (n=17-64)] converging to the ground ionic state S+(4S composite function) formed by vuv excitations of S(3P2,1,0). The converging limits of these Rydberg series have provided more accurate values, 82 985.43+/-0.05, 83 162.94+/-0.05, and 83 559.04+/-0.05 cm(-1) for the respective ionization energies of S(3P0), S(3P1), and S(3P2) to form S+(4S composite function). The relative intensities of the PFI-PI bands for S(3P0), S(3P1), and S(3P2) have been used to determine the branching ratios for these fine structure states, S(3P0):S(3P1):S(3P2)=1.00:1.54:3.55, produced by photodissociation of CS2 at 193.3 nm.

Vacuum ultraviolet excitation spectroscopy of the autoionizing Rydberg states of atomic sulfur in the 73350–84950cm−1 frequency range

The photoionization efficiency (PIE) spectra of metastable sulfur (S) atoms in the 1 D and 1 S states have been recorded in the 73 350-84 950 cm(-1) frequency range by using a velocity-mapped ion imaging apparatus that uses a tunable vacuum ultraviolet laser as the ionization source. The S(1 D) and S(1 S) atoms are produced by the 193 nm photodissociation of CS2. The observed PIE spectra of S(1 D) and S(1 S) shows 35 autoionizing resonances with little or no contribution from direct photoionization into the S+(4S 3/2)+e(-) ionization continuum. Velocity-mapped ion images of the S+ at the individual autoionizing Rydberg resonances are used to distinguish whether the lower state of the resonance originates from the 1 D, 1 S, or 3P states. The analysis and assignment of the Rydberg peaks revealed 22 new Rydberg states that were not previously known. The autoionization lifetimes tau of the Rydberg states are derived from the linewidths by fitting the lines with the Fano formula. Deviations from the scaling law of tau(n*) proportional to, n*3, where n* is the effective quantum number of the Rydberg state, are observed. This observation is ascribed to perturbations by nearby triplet Rydberg states, which shorten the autoionization lifetimes of the singlet Rydberg levels.

Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

We give two constructions of sets of masks on cograssmannian permutations that can be used in Deodhar’s formula for Kazhdan–Lusztig basis elements of the Iwahori–Hecke algebra. The constructions are respectively based on a formula of Lascoux–Schützenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott–Samelson resolution. These constructions give distinct answers to a question of Deodhar.

An Explicit Derivation of the Möbius Function for Bruhat Order

We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma.

Using carry-truncated addition to analyze add-rotate-xor hash algorithms

Abstract. We introduce a truncated addition operation on pairs of N-bit binary numbers that interpolates between ordinary addition mod and bitwise addition in . We use truncated addition to analyze hash functions that are built from the bit operations add, rotate, and xor, such as Blake, Skein, and Cubehash. Any ARX algorithm can be approximated by replacing ordinary addition with truncated addition, and we define a metric on such algorithms which we call the sensitivity. This metric measures the smallest approximation agreeing with the full algorithm a statistically useful portion of the time (we use ). Because truncated addition greatly reduces the complexity of the non-linear operation in ARX algorithms, the approximated algorithms are more susceptible to both collision and pre-image attacks, and we outline a potential collision attack explicitly. We particularize some of these observations to the Skein hash function.

Solitaire Mancala Games and the Chinese Remainder Theorem

Abstract Mancala is a generic name for a family of sowing games that are popular all over the world. There are many two-player mancala games in which a player may move again if their move ends in their own store. In this work, we study a simple solitaire mancala game called Tchoukaillon that facilitates the analysis of “sweep” moves, in which all of the stones on a portion of the board can be collected into the store. We include a self-contained account of prior research on Tchoukaillon, as well as a new description of all winning Tchoukaillon boards with a given length. We also prove an analogue of the Chinese Remainder Theorem for Tchoukaillon boards, and give an algorithm to reconstruct a complete winning Tchoukaillon board from partial information. Finally, we propose a graph-theoretic generalization of Tchoukaillon for further study.