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[0909.5459] On the Generalized Climbing Stairs Problem
Let  be a subset of the positive integers, and M be a positive integer. Mohammad K. Azarian, inspired by work of Tony Colledge, considered the number of ways to climb a staircase containing n stairs using "step-sizes" s∈ and multiplicities at most M.
In this exposition, we find a solution via generating functions, i.e., an expression which counts the number of partitions n=∑s∈mss satisfying 0≤ms≤M. We then use this result to answer a series of questions posed by Azarian, thereby showing a link with ten sequences listed in the On-Line Encyclopedia of Integer Sequences. We conclude by posing open questions which seek to count the number of compositions of n.
combinatorics  enumeration  rather-interesting  counting  nudge-targets  consider:rediscovery  consider:supervised-learning 
6 weeks ago by Vaguery
[1510.07499] An example of geometric origami design with benefit of graph enumeration algorithms
This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from brute force enumeration, up to the equivalence to a dedicated problem in graph theory. This leads to algorithms starting from an untractable case on modern computers, up to a run of few seconds on a portable personal computer. This emphasizes the need for a prior analysis by humans before considering the assistance of computers for complex design problems. The graph problem is an enumeration of spanning trees from a grid graph, leading to a coarse scale description of the topology of the paper edge on the flat-folded state.
computational-geometry  origami  rather-interesting  planning  algorithms  engineering-design  looking-to-see  approximation  combinatorics  counting  to-write-about 
11 weeks ago by Vaguery
Why did Sumerians use the sexagesimal system?
TIL that 60 is countable using the fingers of both hands, and this ancient Mesopotamian counting technique is still used in India, Pakistan, Indochina, Afghanistan, Iran, Turkey, Syria and Egypt apparently
sexagesimal  12  60  counting  fingers  history  sumerian  mesopotamia 
march 2019 by jm
[1701.06377] Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, simple, connected graph. An arithmetical structure on G is a pair of positive integer vectors d,r such that (diag(d)−A)r=0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients (2n−1n−1), and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.
graph-theory  combinatorics  enumeration  counting  to-understand  matrices  feature-construction 
december 2018 by Vaguery
[1808.06313] Binomial coefficients and multifactorial numbers through generative grammars
In this paper, the formal derivative operator defined with respect to context-free grammars is used to prove some properties about binomial coefficients and multifactorial numbers. In addition, we extend the formal derivative operator to matrix grammars and show that multifactorial numbers can also be generated.
combinatorics  strings  rather-interesting  counting  to-write-about 
december 2018 by Vaguery
[1810.04692] Probability distributions related to tilings of non-convex Polygons
This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete tacnode kernel). Here this kernel is used to find the probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts. These distributions are new.
combinatorics  tiling  counting  rather-interesting  phase-transitions  condensed-matter  statistical-mechanics  feature-extraction  representation  to-write-about  consider:feature-discovery 
december 2018 by Vaguery
Khan Academy Kids GIF - Find & Share on GIPHY
GIF kids, bird, cars, children, zoom, learning, educational, numbers, preschool, counting, peck, 12345, vehicles, kindergarten, khankids, early learning, app for kids, khan academy kids, khan kids, early childhood education, count to 5 Giphy ______
kids  bird  cars  children  zoom  learning  educational  numbers  preschool  counting  peck  12345  vehicles  kindergarten  khankids  early  app  for  khan  academy  c 
september 2018 by architektura
How the Words Used For Numbers in Languages Around the World Often Have Anatomical Roots
In a numenary episode of their incredibly informative whiteboard series for Mental Floss, linguist Arika Okrent and illustrator Sean O’Neill verbally and visually explain how in languages around the world, the words that are used for numbers come from easily accessible anatomical sources.

So most cultures didn’t settle on the systems they did because they’re the best, but because of what we happened to have on hand—our fingers, and in the case of base 20, our toes too. How do we know this Sometimes the words themselves tell us. Many number words around the world are etymologically derived from words for hands, fingers, and toes.
numbers  math  counting  base  60 
august 2018 by Quercki
GitHub is where people build software. More than 28 million people use GitHub to discover, fork, and contribute to over 85 million projects.
golang  counting 
august 2018 by geetarista

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