5 research outputs found
Two-stack-sorting with pop stacks
We consider the set of permutations that are sorted after two passes through
a pop stack. We characterize these permutations in terms of forbidden patterns
(classical and barred) and enumerate them according to the ascent statistic.
Then we show these permutations to be in bijection with a special family of
polyominoes. As a consequence, the permutations sortable by this machine are
shown to have the same enumeration as three classical permutation classes.Comment: 18 pages, 7 figure
A polyominoes-permutations injection and tree-like convex polyominoes
AbstractPlane polyominoes are edge-connected sets of cells on the orthogonal lattice Z2, considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence an=6an−1−14an−2+16an−3−9an−4+2an−5, and compute the closed-form formula an=2n+2−(n3−n2+10n+4)/2
Flip-sort and combinatorial aspects of pop-stack sorting
Flip-sort is a natural sorting procedure which raises fascinating
combinatorial questions. It finds its roots in the seminal work of Knuth on
stack-based sorting algorithms and leads to many links with permutation
patterns. We present several structural, enumerative, and algorithmic results
on permutations that need few (resp. many) iterations of this procedure to be
sorted. In particular, we give the shape of the permutations after one
iteration, and characterize several families of permutations related to the
best and worst cases of flip-sort. En passant, we also give some links between
pop-stack sorting, automata, and lattice paths, and introduce several tactics
of bijective proofs which have their own interest.Comment: This v3 just updates the journal reference, according to the
publisher wis
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum