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9085 results in Discrete Mathematics, Information Theory and Coding

Page 86 of 455

  • A Tutte Polynomial for Maps

  • ANDREW GOODALL, THOMAS KRAJEWSKI, GUUS REGTS, LLUÍS VENA
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 6 / November 2018
    Published online by Cambridge University Press:
    12 April 2018, pp. 913-945

  • We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

  • Ewens Sampling and Invariable Generation

  • GERANDY BRITO, CHRISTOPHER FOWLER, MATTHEW JUNGE, AVI LEVY
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 6 / November 2018
    Published online by Cambridge University Press:
    12 April 2018, pp. 853-891

  • We study the number of random permutations needed to invariably generate the symmetric group Sn when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the special case α = 1 corresponds to uniformly random permutations.

    For strong α-logarithmic measures and almost every α, we show that precisely ⌈(1−αlog2)−1⌉ permutations are needed to invariably generate Sn with asymptotically positive probability. A corollary is that for many other probability measures on Sn no fixed number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Łuczak and Bovey to permutations obtained from the Ewens sampling formula.

Page 86 of 455