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Register a new userExistence proofs in combinatorics using independence
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This note is purely expository and is in Russian. We show how to prove interesting combinatorial results using the local Lovasz lemma. The note is accessible for students having basic knowledge of combinatorics; the notion of independence is defined and the Lovasz lemma is stated and proved. Our exposition follows `Probabilistic methods of N. Alon and J. Spencer. The main difference is that we show how the proof could have been invented. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.
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This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and German. We provide elementary proofs for the original convention with ${mathcal B}_1=1/2$ and also for the current convention with $B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic (umbral) notation.
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.
We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.
In probability theory, the independence is a very fundamental concept, but with a little mystery. People can always easily manipulate it logistically but not geometrically, especially when it comes to the independence relationships among more that two variables, which may also involve conditional independence. Here I am particularly interested in visualizing Markov chains which have the well known memoryless property. I am not talking about drawing the transition graph, instead, I will draw all events of the Markov process in a single plot. Here, to simplify the question, this work will only consider dichotomous variables, but all the methods actually can be generalized to arbitrary set of discrete variables.
For every constant c > 0, we show that there is a family {P_{N, c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: * For every family {f_n} of polynomials in VP, where f_n is an n variate polynomial of degree at most n^c with bounded integer coefficients and for N = binom{n^c + n}{n}, P_{N,c} emph{vanishes} on the coefficient vector of f_n. * There exists a family {h_n} of polynomials where h_n is an n variate polynomial of degree at most n^c with bounded integer coefficients such that for N = binom{n^c + n}{n}, P_{N,c} emph{does not vanish} on the coefficient vector of h_n. In other words, there are efficiently computable equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence emph{against} a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017). Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes. Our proofs also extend to finite fields of small size.
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