3,460 research outputs found
Enumerating (2+2)-free posets by indistinguishable elements
A poset is said to be (2+2)-free if it does not contain an induced subposet
that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two
elements in a poset are indistinguishable if they have the same strict up-set
and the same strict down-set. Being indistinguishable defines an equivalence
relation on the elements of the poset. We introduce the statistic maxindist,
the maximum size of a set of indistinguishable elements. We show that, under a
bijection of Bousquet-Melou et al., indistinguishable elements correspond to
letters that belong to the same run in the so-called ascent sequence
corresponding to the poset. We derive the generating function for the number of
(2+2)-free posets with respect to both maxindist and the number of different
strict down-sets of elements in the poset. Moreover, we show that (2+2)-free
posets P with maxindist(P) at most k are in bijection with upper triangular
matrices of nonnegative integers not exceeding k, where each row and each
column contains a nonzero entry. (Here we consider isomorphic posets to be
equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1
correspond to upper triangular binary matrices where each row and column
contains a nonzero entry, and whose entries sum to n. We derive a generating
function counting such matrices, which confirms a conjecture of Jovovic, and we
refine the generating function to count upper triangular matrices consisting of
nonnegative integers not exceeding k and having a nonzero entry in each row and
column. That refined generating function also enumerates (2+2)-free posets
according to maxindist. Finally, we link our enumerative results to certain
restricted permutations and matrices.Comment: 16 page
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
Permutation Classes of Polynomial Growth
A pattern class is a set of permutations closed under the formation of
subpermutations. Such classes can be characterised as those permutations not
involving a particular set of forbidden permutations. A simple collection of
necessary and sufficient conditions on sets of forbidden permutations which
ensure that the associated pattern class is of polynomial growth is determined.
A catalogue of all such sets of forbidden permutations having three or fewer
elements is provided together with bounds on the degrees of the associated
enumerating polynomials.Comment: 17 pages, 4 figure
On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts
Tewodros Amdeberhan and Armin Straub initiated the study of enumerating
subfamilies of the set of (s,t)-core partitions. While the enumeration of
(n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it
equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core
partitions into odd parts remains elusive.
Straub computed the first eleven terms of that sequence, and asked for a
"formula," or at least a fast way, to compute many terms. While we are unable
to find a "fast" algorithm, we did manage to find a "faster" algorithm, which
enabled us to compute 23 terms of this intriguing sequence. We strongly believe
that this sequence has an algebraic generating function, since a "sister
sequence" (see the article), is OEIS sequence A047749 that does have an
algebraic generating function. One of us (DZ) is pledging a donation of 100
dollars to the OEIS, in honor of the first person to generate sufficiently many
terms to conjecture (and prove non-rigorously) an algebraic equation for the
generating function of this sequence, and another 100 dollars for a rigorous
proof of that conjecture.
Finally, we also develop algorithms that find explicit generating functions
for other, more tractable, families of (n+1,n+2)-core partitions.Comment: 12 pages, accompanied by Maple package. This version announces that
our questions were all answered by Paul Johnson, and a donation to the OEIS,
in his honor, has been mad
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