Related papers: Thue, Combinatorics on words, and conjectures insp…
A conjecture of Chunwei Song on a limiting case of the q,t-Schr\"{o}der theorem is proved combinatorially. The proof matches pairs of tableaux to Catalan words in a manner that preserves differences in the maj statistic.
Sierpinski's triangle is a fractal and the Prouhet-Thue-Morse word is sufficiently chaotic to avoid cubes. Here we observe that there is at least a tenuous connection between them: the Sierpinski triangle is evident in Pascal's triangle mod…
Let ftm = 0111010010001... be the analogue of the Thue-Morse sequence in Fibonacci representation. In this note we show how, using the Walnut theorem-prover, to obtain a measure of its complexity, previously studied by Jamet, Popoli, and…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give direct proofs…
By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and then we obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace…
Inspired by recent work of I. Baoulina, we give a simultaneous generalization of the theorems of Chevalley-Warning and Morlaye.
We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We…
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by…
We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim.
In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding…
Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod 2), where s_2 is the sum of the bits in the binary representation of n. It is well known that for any integer k>=1 the frequency of the letter "1" in the…
The main result of this announcement is a formula for the tensor product of the class of a homogeneous line bundle with a Schubert class, expressed as a K(X)-linear combination of Schubert classes. We believe that this formula is the most…
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and…
We provide an explicit formula for the toric $h$-contribution of each cubical shelling component, and a new combinatorial model to prove Clara Chan's result on the non-negativity of these contributions. Our model allows for a variant of the…
Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a…
We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group $\mu_\ell$ of $\ell$-th roots of unity, where $\ell$ is any prime. To achieve this we develop…
S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.