Let's reinvent subtraction
Subtraction is a powerful technique for creating new bijections from old. Let's reinvent it! While we're at it, let's reinvent division as well.
math.CO↗
arXiv subjects
Peter G. Doyle
Publications and source records attributed to Peter G. Doyle.
Conway's drum quilts
A `transplantable pair' is a pair of glueing diagrams that can be used to create pairs of plane domains that are isospectral for the Laplace operator. We present a host of transplantable pairs worked out by John Conway using his theory of quilts
math.GT↗
A category for bijective combinatorics
The category of matchings between finite sets extends to the category of cobordisms of signed sets. A chain of cobordisms that starts and ends with unsigned sets A and B yields a matching from A to B. This is a convenient way to package the involution principle of Garsia and Milne, which reveals itself to have little to do with involutions.
math.CO↗
Frobenius's last proof
Around about 1917, Issai Schur rediscovered the Rogers-Ramanujan identities, and proved a system of polynomial identities that imply them. Schur wrote that Georg Frobenius (his former advisor) had shown him a simple, direct proof of these polynomial identities. Schur did not see fit to reveal Frobenius's proof, preferring his own rather complicated proof. But it is easy enough to guess what this `simple, direct' proof must have been. As Frobenius died in 1917, we may call this `Frobenius's last proof'.
math.HO↗
Geometry and the Imagination in Minneapolis
This document consists of the collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991. The workshop was based on a course `Geometry and the Imagination' which we had taught twice before at Princeton.
math.DG↗
Spectral invariants and playing hide-and-seek on surfaces
We prove the expected duration of a game of hide-and-seek played on a Riemannian manifold under the laws of Brownian Motion is a spectral invariant: it is a zeta-regularized version of the `trace' of the Laplacian. An analogous hide-and-seek game may be played on Markov chains, where the spectral invariant that emerges is a classical quantity known as Kemeny's constant. We develop the analogies between the two settings in order to highlight the connections between the regularized trace and Kemeny's constant. Our proof relies on the connections between Green's functions and expected hitting times, and the fact that the regularized trace may be approached via the Green's function.
math.SP↗
Commuting time geometry of ergodic Markov chains
We show how to map the states of an ergodic Markov chain to Euclidean space so that the squared distance between states is the expected commuting time. We find a minimax characterization of commuting times, and from this we get monotonicity of commuting times with respect to equilibrium transition rates. All of these results are familiar in the case of time-reversible chains, where techniques of classical electrical theory apply. In presenting these results, we take the opportunity to develop Markov chain theory in a `conformally correct' way
math.PR↗
Equivariant division
Let C be a non-empty finite set, and Gamma a subgroup of the symmetric group S(C). Given a bijection f:A cross C to B cross C, the problem of Gamma-equivariant division is to find a quotient bijection h:A to B respecting whatever symmetries f may have under the action of S(A) cross S(B) cross Gamma. Say that Gamma is fully cancelling if this is possible for any f, and finitely cancelling if it is possible providing A,B are finite. Feldman and Propp showed that a permutation group is finitely cancelling just if it has a globally fixed point. We show here that a permutation group is fully cancelling just if it is trivial. This sheds light on the fact that all known division algorithms that eschew the Axiom of Choice depend on fixing an ordering for the elements of C.
math.CO↗
Division by four
Write A<=B if there is an injection from A to B, and A==B if there is a bijection. We give a simple proof that for finite n, nA<=nB implies A<=B. From the Cantor-Bernstein theorem it then follows that nA==nB implies A==B. These results have a long and tangled history, of which this paper is meant to be the culmination.
math.LO↗
Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent
Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; and the number of primitive closed geodesics of each length and orientability class, counting a geodesic running along the boundary as half orientation-preserving and half orientation-reversing, and discounting imprimitive geodesics appropriately. This implies that Laplace-isospectral hyperbolic 2-orbifolds determine equivalent linear representations of Isom(H^2), and are isospectral for any natural operator.
math.DG↗
Cyclic groups with the same Hodge series
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms for all p, but not for all natural operators. Here we explain, simplify, and extend their investigations
math.RA↗
Stackable and queueable permutations
There is a natural bijection between permutations obtainable using a stack (those avoiding the pattern 312) and permutations obtainable using a queue (those avoiding 321). This bijection is equivalent to one described by Simion and Schmidt in 1985. We argue that this bijection might well have been found back in 1968 by readers of volume 1 of Knuth's *The Art of Computer Programming*, if Knuth had not assigned difficulty ratings to his exercises.
math.CO↗
Maybe there's no such thing as a random sequence
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it would seem that the set of random sequences should have full measure. But in fact there might be no random sequences, because for all we know, there might be no undefinable sets.
math.PR↗
The Kemeny constant of a Markov chain
Given an ergodic finite-state Markov chain, let M_{iw} denote the mean time from i to equilibrium, meaning the expected time, starting from i, to arrive at a state selected randomly according to the equilibrium measure w of the chain. John Kemeny observed that M_{iw} does not depend on starting the point i. The common value K=M_{iw} is the Kemeny constant or seek time of the chain. K is a spectral invariant, to wit, the trace of the resolvent matrix. We review basic facts about the seek time, and connect it to the bus paradox and the Central Limit Theorem for ergodic Markov chains.
math.PR↗
Isospectral hyperbolic surfaces have matching geodesics
We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.
math.DG↗
Frustration solitaire
In this expository article, we discuss the rank-derangement problem, which asks for the number of permutations of a deck of cards such that each card is replaced by a card of a different rank. This combinatorial problem arises in computing the probability of winning the game of `frustration solitaire'. The solution is a prime example of the method of inclusion and exclusion. We also discuss and announce the solution to Montmort's `Probleme du Treize', a related problem dating back to circa 1708.
math.PR↗
Riffles, ruffles, and the turning algebra
The rising algebra is a subalgebra of the group algebra of the symmetric group S_n, gotten by lumping together permutations having the same number of rising sequences. This well-known algebra arises naturally when studying riffle shuffles. Here we introduce a number of other subalgebras that arise naturally when studying `ruffles', which are like riffles except that after cutting the deck you turn over the bunch of cards that were on the bottom. This orphaned draft offers no context or motivation, and uses idiosyncratic notation and terminology that `seemed like a good idea at the time'. We're making it available because it has been cited in this form.
math.CO↗
A 27-vertex graph that is vertex-transitive and edge-transitive but not l-transitive
I describe a 27-vertex graph that is vertex-transitive and edge-transitive but not 1-transitive. Thus while all vertices and edges of this graph are similar, there are no edge-reversing automorphisms.
math.CO↗