arXiv subjects

Ron Graham

Publications and source records attributed to Ron Graham.

At least 19 recordsLinked to original sources

arXiv2020

Guessing about Guessing: Practical Strategies for Card Guessing with Feedback

In simple card games, cards are dealt one at a time and the player guesses each card sequentially. We study problems where feedback (e.g. correct/incorrect) is given after each guess. For decks with repeated values (as in blackjack where suits do not matter) the optimal strategy differs from the "greedy strategy" (of guessing a most likely card each round). Further, both optimal and greedy strategies are far too complicated for real time use by human players. Our main results show that simple heuristics perform close to optimal.

math.PR↗

arXiv2020

Card Guessing with Partial Feedback

Consider the following experiment: a deck with $m$ copies of $n$ different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of "feedback" is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in $n$, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

math.PR↗

arXiv2019

Slow Fibonacci Walks

For a positive integer $n$, we study the number of steps to reach $n$ by a {\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $\phi n$ where $\phi=(1+\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$. We give explicit formulas for the choice of the starting pairs and the determination of $s(n)$ by giving a characterization theorem. We also derive a number of density results concerning the distribution of down and up cases (that is, those $n$ with $a_{s+1}=\lfloor \phi n\rfloor$ or $\lceil \phi n \rceil$, respectively), as well as for more general `paradoxical' cases.

math.NT↗

arXiv2015

Juggling card sequences

Juggling patterns can be described by a sequence of cards which keep track of the relative order of the balls at each step. This interpretation has many algebraic and combinatorial properties, with connections to Stirling numbers, Dyck paths, Narayana numbers, boson normal ordering, arc-labeled digraphs, and more. Some of these connections are investigated with a particular focus on enumerating juggling patterns satisfying certain ordering constraints, including where the number of crossings is fixed.

math.CO↗

arXiv2015

Coefficients of the Inflated Eulerian Polynomial

It follows from work of Chung and Graham that for a certain family of polynomials $T_{n}(x)$, derived from the descent statistic on permutations, the coefficient sequence of $T_{n-1}(x)$ coincides with that of the polynomial $T_{n}(x)/\left(1+x+\cdots+x^{n-1}\right)$. We observed computationally that the inflated $\mathbf{s}$-Eulerian polynomial $Q_{n}^{(\mathbf{s})}(x)$, which satisfies $Q_{n}^{(\mathbf{s})}(x) = T_{n}(x)$ when $\mathbf{s}=(1,2,\ldots,n)$, also satisfies this property for many sequences $\mathbf{s}$. In this work we characterize those sequences $\mathbf{s}$ for which the coefficient sequence of $Q_{n-1}^{(\mathbf{s})}(x)$ coincides with that of the polynomial $Q_{n}^{(\mathbf{s})}(x)/\left(1+x+\cdots+x^{s_{n}-1}\right)$. In particular, we show that all nondecreasing sequences satisfy this property. We also settle a conjecture of Pensyl and Savage by showing that the inflated $\mathbf{s}$-Eulerian polynomials are unimodal for all choices of positive integer sequences ${\bf s}$. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, K\"oppe, Savage, and Zafeirakopoulos.

math.NT↗

arXiv2015

Partition and sum is fast

We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller number and repeated application can always reduce the number to a single digit. We show that surprisingly few iterations of this operation are needed to get down to a single digit.

math.HO↗

arXiv2014

The mathematics of the flip and horseshoe shuffles

We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is "reversed", and then the cards are interlaced. Flip shuffles are when the reversal comes from flipping the half over so that we also need to account for face-up/face-down configurations while horseshoe shuffles are when the order of the cards are reversed but all cards still face the same direction. We show that these shuffles are closely related to faro shuffling and determine the order of the associated shuffling groups.

math.CO↗

arXiv2014

On Levine's notorious hat puzzle

The Levine hat game requires $n$ players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with $n = 2$ players (and a conjecture for an optimal strategy in this case). We also prove that $V_n$, the optimal value of the joint success probability in the $n$-player game, is a strictly decreasing function of $n$.

math.CO↗

arXiv2013

Unseparated pairs and fixed points in random permutations

In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion principle, the introduction of two different Markov chains to generate uniform random permutations, and the construction of a combinatorial bijection. We also obtain the distribution of the analogous set for circular permutations that consists of those k in [n] such that \Pi(k+1 mod n) = \Pi(k) + 1 mod n. This latter random set is just the set of fixed points of the commutator [\rho, \Pi], where \rho is the n-cycle (1,2,...,n). We show for a general permutation \eta that, under weak conditions on the number of fixed points and 2-cycles of \eta, the total variation distance between the distribution of the number of fixed points of [\eta,\Pi] and a Poisson distribution with expected value 1 is small when n is large.

math.PR↗

arXiv2012

Unrolling residues to avoid progressions

We consider the problem of coloring $[n]={1,2,...,n}$ with $r$ colors to minimize the number of monochromatic $k$ term arithmetic progressions (or $k$-APs for short). We show how to extend colorings of $\mathbb{Z}_m$ which avoid nontrivial $k$-APs to colorings of $[n]$ by an unrolling process. In particular, by using residues to color $\mathbb{Z}_m$ we produce the best known colorings for minimizing the number of monochromatic $k$-APs for coloring with $r$ colors for several small values of $r$ and $k$.

math.CO↗

arXiv2010

Origami rings

Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let $L_\alpha(p)$ be the line in the complex plane through $p$ with angle $\alpha$ (with respect to the real axis). Given a fixed collection $U$ of angles, let $\RU$ be the points that can be obtained by starting with $0$ and $1$, and then recursively adding intersection points of the form $L_\alpha(p) \cap L_\beta(q)$, where $p, q$ have been constructed already, and $\alpha, \beta$ are distinct angles in $U$. Our main result is that if $U$ is a group with at least three elements, then $\RU$ is a subring of the complex plane, i.e., it is closed under complex addition and multiplication. This enables us to answer a specific question about origami folds: if $n \ge 3$ and the allowable angles are the $n$ equally spaced angles $k\pi/n$, $0 \le k < n$, then $\RU$ is the ring $\Z[\zeta_n]$ if $n$ is prime, and the ring $\Z[1/n,\zeta_{n}]$ if $n$ is not prime, where $\zeta_n := \exp(2\pi i/n)$ is a primitive $n$-th root of unity.

math.CO↗

arXiv2010

On minimal colorings without monochromatic solutions to a linear equation

For a ring R and system L of linear homogeneous equations, we call a coloring of the nonzero elements of R minimal for L if there are no monochromatic solutions to L and the coloring uses as few colors as possible. For a rational number q and positive integer n, let E(q,n) denote the equation $\sum_{i=0}^{n-2} q^{i}x_i = q^{n-1}x_{n-1}$. We classify the minimal colorings of the nonzero rational numbers for each of the equations E(q,3) with q in {3/2,2,3,4}, for E(2,n) with n in {3,4,5,6}, and for x_1+x_2+x_3=4x_4. These results lead to several open problems and conjectures on minimal colorings.

math.CO↗

arXiv2010

Subdivision by bisectors is dense in the space of all triangles

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) any given triangle when the point that we use to subdivide is the incenter. We also show that the smallest angle in a "typical" triangle after repeated subdivision for many generations does not have the smallest angle going to zero.

math.CO↗

arXiv2010

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."

math.CO↗

arXiv2010

Shuffling with ordered cards

We consider a problem of shuffling a deck of cards with ordered labels. Namely we split the deck of N=k^tq cards (where t>=1 is maximal) into k equally sized stacks and then take the top card off of each stack and sort them by the order of their labels and add them to the shuffled stack. We show how to find stacks of cards invariant and periodic under the shuffling. We also show when gcd(q,k)=1 the possible periods of this shuffling are all divisors of order_k(N-q).

math.CO↗

arXiv2010

Descent polynomials for permutations with bounded drop size

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.

math.CO↗

arXiv2008

Can you hear the shape of a Beatty sequence?

Let K(x_1,...,x_d) be a polynomial. If you are not given the real numbers \alpha_1, \alpha_2, ...,\alpha_d, but are given the polynomial K and the sequence a_n=K(\floor{n\alpha_1},\floor{n\alpha_2},...,\floor{n\alpha_d}), can you deduce the values of \alpha_i? Not, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing \alpha_i from the integer sequence with nested flooring (\floor{\floor{... \floor{\floor{n\alpha_1}\alpha_2}... \alpha_{d-1}}\alpha_d})_{n=1}^\infty.

math.NT↗

arXiv2008

Jumping sequences

An integer sequence a(n) is called a jump sequence if a(1)=1 and 1<=a(n) =2. Such a sequence has the property that a^k(n)=a(a(...(a(n))...)) goes to 1 in finitely many steps and we call the pattern (n,a(n),a^2(n),...,a^k(n)=1) a jumping pattern from n down to 1. In this paper we look at jumping sequences which are weight minimizing with respect to various weight functions (where a weight w(i,j) is given to each jump from j down to i). Our main result is to show that if w(i,j)=(i+j)/i^2 then the cost minimizing jump sequence has the property that the number m satisfies m=a^q(p) for arbitrary q and some p (depending on q) if and only if m is a Pell number.

math.CO↗