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Full-Text Articles in Physical Sciences and Mathematics
Almost Avoiding Permutations, Robert Brignall, Shalosh B. Ekhad, Rebecca Smith, Vincent Vatter
Almost Avoiding Permutations, Robert Brignall, Shalosh B. Ekhad, Rebecca Smith, Vincent Vatter
Dartmouth Scholarship
The permutation π of length n, written in one-line notation as π (1)π (2)· · · π (n), is said to contain the permutation σ if π has a subsequence that is order isomorphic to σ, and each such subsequence is said to be an occurrence of σ in π or simply a σ pattern. For example, π = 491867532 contains σ = 51342 because of the subsequence π (2)π (3)π (5)π (6)π (9) = 91672. Permutation containment is easily seen to be a partial order on the set of all (finite) permutations, which we simply denote by ≤. If …
Submodular Percolation, Graham R. Brightwell, Peter Winkler
Submodular Percolation, Graham R. Brightwell, Peter Winkler
Dartmouth Scholarship
Let $f:{\cal L}\to\mathbb{R}$ be a submodular function on a modular lattice ${\cal L}$; we show that there is a maximal chain ${\cal C}$ in ${\cal L}$ on which the sequence of values of f is minimal among all paths from 0 to 1 in the Hasse diagram of ${\cal L}$, in a certain well-behaved partial order on sequences of reals. One consequence is that the maximum value of f on ${\cal C}$ is minimized over all such paths—i.e., if one can percolate from 0 to 1 on lattice points X with $f(X)\le b$, then one can do so along a …
The Number Of Permutations Realized By A Shift, Sergi Elizalde
The Number Of Permutations Realized By A Shift, Sergi Elizalde
Dartmouth Scholarship
A permutation $\pi$ is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J. M. Amigó, S. Elizalde, and M. B. Kennel, J. Combin. Theory Ser. A, 115 (2008), pp. 485–504] that the shortest forbidden patterns of the shift on N symbols have length $N+2$. In this paper we give …