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In 1998, Allouche, Peyri`{e}re, Wen and Wen showed that the Hankel determinant $H_n$ of the Thue-Morse sequence over ${-1,1}$ satisfies $H_n/2^{n-1}equiv 1~(mathrm{mod}~2)$ for all $ngeq 1$. Inspired by this result, Fu and Han introduced emph{apwenian} sequences over ${-1,1}$, namely, $pm 1$ sequences whose Hankel determinants satisfy $H_n/2^{n-1}equiv 1~(mathrm{mod}~2)$ for all $ngeq 1$, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let $f(z)$ be the generating functions of such apwenian sequences. We show that for all integer $bge 2$ with $f(1/b) eq 0$, the real number $f(1/b)$ is transcendental and its irrationality exponent is equal to $2$. Besides, we also derive a criterion for zero-one apwenian sequences whose Hankel determinants satisfy $H_nequiv 1~(mathrm{mod}~2)$ for all $ngeq 1$. We find that the only zero-one apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also given. Furthermore, we prove that all Sturmian sequences over ${-1,1}$ or ${0,1}$ are not apwenian. And we conjecture that fixed points of substitution of non-constant length over ${-1,1}$ or ${0,1}$ can not be apwenian.
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Allouche and Shallit introduced the notion of a regular power series as a generalization of automatic sequences. Becker showed that all regular power series satisfy Mahler equations and conjectured equivalent conditions for the converse to be true. We prove a stronger form of Beckers conjecture for a subclass of Mahler power series.
Let $kappa$ be a positive real number and $minmathbb{N}cup{infty}$ be given. Let $p_{kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(lfloor ell^{kappa}rfloor)_{ellin mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{kappa, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.
Given a finite set of nonnegative integers A with no 3-term arithmetic progressions, the Stanley sequence generated by A, denoted S(A), is the infinite set created by beginning with A and then greedily including strictly larger integers which do not introduce a 3-term arithmetic progressions in S(A). Erdos et al. asked whether the counting function, S(A,x), of a Stanley sequence S(A) satisfies S(A,x)>x^{1/2-epsilon} for every epsilon>0 and x>x_0(epsilon,A). In this paper we answer this question in the affirmative; in fact, we prove the slightly stronger result that S(A,x)geq (sqrt{2}-epsilon)sqrt{x} for xgeq x_0(epsilon,A).
The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice.
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. The purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.
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