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Young module multiplicities and classifying the indecomposable Young permutation modules
We study the multiplicities of Young modules as direct summands of
permutation modules on cosets of Young subgroups. Such multiplicities have
become known as the p-Kostka numbers. We classify the indecomposable Young
permutation modules, and, applying the Brauer construction for p-permutation
modules, we give some new reductions for p-Kostka numbers. In particular we
prove that p-Kostka numbers are preserved under multiplying partitions by p,
and strengthen a known reduction given by Henke, corresponding to adding
multiples of a p-power to the first row of a partition.Comment: 22 page
Fantastic Patterns and Where Not to Find Them
Interesting patterns are everywhere we look, but what happens when we try to avoid patterns? A permutation is a list of numbers in a specific order. When we avoid a pattern, we try not to order those numbers in certain ways. For example, the permutation 45312 avoids the 123 pattern because no three elements in the permutation are in an increasing order. In our work, we studied the permutations that avoid two different patterns of length three. We focused on the distribution of peaks, valleys, double ascents, and double descents over these sets of permutations
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
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