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The ith cycle minor of a permutation p of the set {1, 2, . . . , n} is the permutation formed by deleting an entry i from the decomposition of p into disjoint cycles and reducing each remaining entry larger than i by 1. In this paper, we show that any permutation of {1, 2, . . . , n} can be reconstructed from its set of cycle minors if and only if n ≥ 6. We(More)
All continuous endomorphisms f∞ of the shift dynamical system S on the 2-adic integers Z2 are induced by some f : Bn → {0, 1}, where n is a positive integer, Bn is the set of n-blocks over {0, 1}, and f∞ (x) = y0y1y2 . . . where for all i ∈ N, yi = f (xixi+1 . . . xi+n−1). Define D : Z2 → Z2 to be the endomorphism of S induced by the map {(00, 0) , (01, 1)(More)
Abstract. Let R(w; q) be Dyson’s generating function for partition ranks. For roots of unity ζ 6= 1, it is known that R(ζ; q) and R(ζ; 1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w; q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w; q)(More)
Given a partition λ of n, a k-minor of λ is a partition of n − k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k ≤ g(n). In particular, partitions of n ≥ k2 + 2k are uniquely determined by their sets of k-minors. This result completely(More)
The 3x+ 1 Conjecture asserts that the T -orbit of every positive integer contains 1, where T maps x 7→ x/2 for x even and x 7→ (3x + 1)/2 for x odd. A set S of positive integers is sufficient if the orbit of each positive integer intersects the orbit of some member of S. In [9] it was shown that every arithmetic sequence is sufficient. In this paper we(More)
Let A be a finite alphabet and let L ⊂ (A∗)n be an n-variable language over A. We say that L is regular if it is the language accepted by a synchronous n-tape finite state automaton, it is quasi-regular if it is accepted by an asynchronous n-tape automaton, and it is weakly regular if it is accepted by a non-deterministic asynchronous n-tape automaton. We(More)
Let Q(n) denote the number of partitions of n into distinct parts. We show that Dyson’s rank provides a combinatorial interpretation of the well-known fact that Q(n) is almost always divisible by 4. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan’s mock theta functions,(More)
The dual of an algebraic curve C in RP defined by the polynomial equation f(x, y, z) = 0 is the locus of points ( ∂f ∂x (a, b, c) : ∂f ∂y (a, b, c) : ∂f ∂z (a, b, c) ) where (a : b : c) ∈ C. The dual can alternatively be defined geometrically as the image under reciprocation of the envelope of tangent lines to the curve. It is known that the dual of an(More)
Let A be a finite alphabet and let L ⊂ (A∗)n be an n-variable language over A. We say that L is regular if it is the language accepted by a synchronous n-tape finite state automaton, it is quasi-regular if it is accepted by an asynchronous n-tape automaton, and it is weakly regular if it is accepted by a non-deterministic asynchronous n-tape automaton. We(More)
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