Enumeration of 3-letter patterns in compositions

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer Conferenc

Building Nim

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture

Supersymmetry on Jacobstahl lattices

It is shown that the construction of Yang and Fendley (2004 {\it J. Phys. A: Math.Gen. {\bf 37}} 8937) to obtainsupersymmetric systems, leads not to the open XXZ chain with anisotropy $\Delta =-{1/2}$ but to systems having dimensions given by Jacobstahl sequences.For each system the ground state is unique. The continuum limit of the spectra of the Jacobstahl systems coincide, up to degeneracies, with that of the $U_q(sl(2))$ invariant XXZ chain for $q=\exp(i\pi/3)$. The relation between the Jacobstahl systems and the open XXZ chain is explained.Comment: 6 pages, 0 figure

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated author email addresse

Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2

We show that from the spectra of the U_q (sl(2)) symmetric XXZ spin-1/2 finite quantum chain at Delta=-1/2 (q=e^{pi i/3}) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Delta=0 (q=e^{pi i/2}). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.Comment: 29 page

Spin Labeling with Nitroxide, Trityl, and Copper Labels for Protein Structure Elucidation by EPR Spectroscopy

In this thesis, the intricate challenges that emerge in the application of site-directed spin labeling (SDSL) techniques for structural investigation of biomolecules are addressed, with a specific focus on nitroxide, trityl and copper spin labels. The primary objective revolves around standardizing procedures to enhance the reliability and reproducibility of these SDSL techniques. Furthermore, the application scope of the different spin labels for pulsed dipolar EPR spectroscopy is discussed. In the first section of this thesis, through a comprehensive SDSL strategy utilizing the Yersinia outer protein O (YopO) as a model system, a set of standardized guidelines for SDSL of proteins employing the widely utilized MTSL spin label is developed. Designed as a multi-laboratory benchmark test, the reproducibility and robustness of data acquisition and analysis on the spin-labeled proteins are evaluated and discussed. In the second section of this thesis, a reliable and reproducible spin-labeling protocol for proteins using trityl spin labels is developed. Through meticulous adjustments and fine-tuning of the labeling conditions, the developed protocol sufficiently suppresses aggregation and over labeling of the proteins and enables site-selective spin labeling using maleimide-functionalized trityl spin labels. Subsequently, the trityl-labeled proteins are compared with regards to their EPR sensitivity and the width of the PDS-derived distance distributions. Furthermore, the feasibility of EPR distance measurements at nanomolar concentrations and within cellular systems is assessed. In the first two sections, ambiguous distance distributions were obtained using both nitroxide and trityl spin labels which suggested two distinct conformations of YopO's a-helical backbone. Therefore, in the third section of this thesis, the conformationally restricted bipedal double histidine motif loaded with paramagnetic copper(II) nitrilotriacetic acid (dHis-Cu2+(NTA)) was employed to distinguish between label and protein conformations. Through its reduced conformational flexibility, it was revealed that the a-helical backbone of YopO adopts a single conformation in solution. The herein presented results provide valuable guidelines to the EPR community as well as non-experts for the application of nitroxide spin labels and PDS-EPR in structural biology, outline a reliable protocol for the routine application of maleimide-functionalized trityl spin labels in PDS-EPR, and showcases an approach to differentiate between spin label and protein conformations using the dHis-Cu2+(NTA) spin label