Partition (number theory)

Summary

In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: :4 :3 + 1 :2 + 2 :2 + 1 + 1 :1 + 1 + 1 + 1 The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagram

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In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.

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Forbidden induced subposets of given height

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Let P be a partially ordered set. The function La* (n, P) denotes the size of the largest family F subset of 2([n]) that does not contain an induced copy of P. It was proved by Methuku and Palvolgyi that there exists a constant C-P (depending only on P) such that La*(n,P) < C-P(left perpendicular n/2 right perpendicular n). However, the order of the constant C-P following from their proof is typically exponential in vertical bar P vertical bar. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer h there exists a constant c(h) such that if P has height at most h, then La* (n, P) = 2 vertical bar P vertical bar, then vertical bar F vertical bar

ACADEMIC PRESS INC ELSEVIER SCIENCE2019

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EPFL2017