Refining Blecher and Knopfmacher's Integer Partition Fixed Points

Recently, Blecher and Knopfmacher explored the notion of fixed points in integer partitions. Here, we distinguish partitions with a fixed point by which value is fixed and analyze the resulting triangle of integers. In particular, we confirm various identities for diagonal sums, row sums, and antidiagonal sums (which are finite for this triangle) and establish a four-term recurrence for triangle entries analogous to Pascal's lemma for the triangle of binomial coefficients. The partition statistics crank and mex arise. All proofs are combinatorial.Comment: 7 pages, 3 figures, 3 table

Development of Action and the Clinical Continuum

The development of action is depicted as consisting of changes in the task-specific couplings between perception, movement, and posture. It is argued that this approach may provide a much needed basis from which attempts can be made at theoretically unifying the constituents of the clinical continuum (viz., early detection, diagnosis, prognosis, and intervention). Illustrative examples germane to this approach are given with regard to how posture serves as a constraint on the emergence of reaching movements and how cortical development influences the coordination of leg movements as revealed by a study involving infants with white matter lesions. Particular attention is paid to early detection and it is recommended that further improvements to this aspect of the clinical continuum can be derived from combining serial qualitative and quantitative (kinematic) assessments with brain-imaging techniques. It is emphasized that quantitative assessments should incorporate experimental manipulations of perception, movement or posture during transitional periods in development. Concluding comments include consideration of the timing of early intervention

Ties in Worst-Case Analysis of the Euclidean Algorithm

We determine all pairs of positive integers below a given bound that require the most division steps in the Euclidean algorithm. Also, we find asymptotic probabilities for a unique maximal pair or an even number of them. Our primary tools are continuant polynomials and the Zeckendorf representation using Fibonacci numbers

Arndt and De Morgan Integer Compositions

In 2013, Joerg Arndt recorded that the Fibonacci numbers count integer compositions where the first part is greater than the second, the third part is greater than the fourth, etc. We provide a new combinatorial proof that verifies his observation using compositions with only odd parts as studied by De Morgan. We generalize the descent condition to establish families of recurrence relations related to two types of compositions: those made of any odd part and certain even parts, and those made of any even part and certain odd parts. These generalizations connect to compositions studied by Andrews and Viennot. New tools used in the combinatorial proofs include two permutations of compositions and a statistic based on the signed pairwise difference between parts.Comment: 13 pages, 1 figure, 11 table

Combinatorics of Multicompositions

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatorial properties of these multicompositions. In particular, we determine the count of multicompositions by number of all parts, number of positive parts, and number of zeros. Then, working from three types of compositions with restricted parts that are counted by the Fibonacci sequence, we find the sequences counting multicompositions with analogous restrictions. With these tools, we give combinatorial proofs of summation formulas for generalizations of the Jacobsthal and Pell sequences.Comment: 13 page

Chemotaxis in uncertain environments: hedging bets with multiple receptor types

Eukaryotic cells are able to sense chemical gradients in a wide range of environments. We show that, if a cell is exposed to a highly variable environment, it may gain chemotactic accuracy by expressing multiple receptor types with varying affinities for the same signal, as found commonly in chemotaxing cells like Dictyostelium. As environment uncertainty is increased, there is a transition between cells preferring a single receptor type and a mixture of types - hedging their bets against the possibility of an unfavorable environment. We predict the optimal receptor affinities given a particular environment. In chemotaxing, cells may also integrate their measurement over time. Surprisingly, time-integration with multiple receptor types is qualitatively different from gradient sensing by a single type -- cells may extract orders of magnitude more chemotactic information than expected by naive time integration. Our results show when cells should express multiple receptor types to chemotax, and how cells can efficiently interpret the data from these receptors

Interferometric Studies of the extreme binary, ϵ\epsilon Aurigae: Pre-eclipse Observations

We report new and archival K-band interferometric uniform disk diameters obtained with the Palomar Testbed Interferometer for the eclipsing binary star $\epsilon$ Aurigae, in advance of the start of its eclipse in 2009. The observations were inteded to test whether low amplitude variations in the system are connected with the F supergiant star (primary), or with the intersystem material connecting the star with the enormous dark disk (secondary) inferred to cause the eclipses. Cepheid-like radial pulsations of the F star are not detected, nor do we find evidence for proposed 6% per decade shrinkage of the F star. The measured 2.27 +/- 0.11 milli-arcsecond K band diameter is consistent with a 300 times solar radius F supergiant star at the Hipparcos distance of 625 pc. These results provide an improved context for observations during the 2009-2011 eclipse.Comment: Accepted for Ap.J. Letters, Oct. 200