Báo cáo toán học: "Self-describing sequences and the Catalan family tree" pps

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... [8].5OpenProblemsThe concepts of stapled sequences and stapling coverings introduced and discussed above lead to some unsolved problems, as follows.1. What is the exact relationship of cutoff and expense? Can ... fact, as shown below, stapling coverings exist for all N ≥ 17.4 Efficient Stapling Coverings An interesting characteristic of stapling covering is the ratio of the number| I | of primes used for ... ) and an algorithm that allows us to transformastaplingcoveringofagivenexpenseintoastaplingcoveringwiththecutoff u(T )=f(ε)?2. Do there exist constructions for efficient stapling coverings of...

... 2-matroidsLet Q1 and Q2be 2-matroids. When Q1 and Q2have the same ground-set and skew classes we shall use Q1+ Q2to denote the parity system which is the sum of Q1 and Q2. When Q1 and Q2have ... sincer1(A) ≤ rt1(A) ≤ r1(A) + 1 and r1(A) ≤ r˜t1(A) ≤ r1(A) + 1. Parity Systems and the Delta-Matroid Intersection ProblemAndr´e Bouchet∗ and Bill Jackson†Submitted: February ... Find whether G1 and G2have orthogonal Euler tours.If G1 and G2are equal to the graph G of Problem 1.7 and each directedtransition of G has the same colour in G1 and G2, then T1 and T2are...

... trees, and therefore a unique bijection between tournament sequences and Meeussen sequences which respects the ideas of extending or truncating a sequence.To better understand Meeussen sequences, ... Wouter Meeussen [private communication, 1999], and we here name sequences with this property Meeussen sequences in his honor. Part of their definitionis similar to that of so-called regular sequences ... Generating functions for generating trees. Preprint, 2000. Tournament Sequences and Meeussen Sequences Matthew CookComputational and Neural Systems ProgramCalifornia Institute of TechnologyPasadena,...

... asymptoticformula for fnkfrom theorem 1.1, together with standard results from randommatrix theory relating the behavior of the eigenvalues of random unitary matricesto the behavior of the eigenvalues of ... restricted increasing subsequence length.Section 1 states and proves the connection between the classical increasing sub-sequence problem and the unitary group. Section 2 extends this to other increasing subsequence ... D1be the following operation on tableaux: delete the top-left element of the tableaux, then move the lesser of the element immediately below and the elementimmediately to the right into the vacated...

... between the self-describing sequences and the Catalan family tree.Theorem 2. The full names of the members of the Catalan family are precisely the self-describing sequences. In other words, they ... proof ofTheorem 3 and the reason is that there is an important difference in the distribution oflabels in the Catalan family tree and the tree of the sequences with unit increase.Theorem 4. ... statements of the theorem are implied by Theorem 1 and the following lemma.Lemma 1. If a is a member of the Catalan family then a = δa. Otherwise, a<δa.Proof. The proof is by induction on the generation...

... #R18 4 Monomial nonnegativity and the Bruhat orderBrian Drake, Sean Gerrish, Mark SkanderaDept. of Mathematics, Brandeis UniversityMS 050, P.O. Box 9110, Waltham, MA 02454bdrake@math.brandeis.eduDept. ... that the coefficient of m1sin the monomial expansion of h hàis negative. the electronic journal of combinatorics 11(2) (2005), #R18 3 Note that the ratio of the coefficients of m1sin the monomial ... paper [5], is the following list of nonnegativity criteria with which one may define the Bruhat order.Theorem 2 Let π and σ be permutations in Sn. The following conditions on π and σare equivalent.1....

... alternations either with blocks of the form π(k,+) for +>or of the form π(k,−) for −<, and horizontal alternations with blocks of the form π(k+,) for k+>kor of the form π(k−,) for ... than the nth Fibonacci number for some n .The definition of sub-2n−1is analogous. the electronic journal of combinatorics 13 (2006), #R54 9 Grid classes and the Fibonacci dichotomy for restricted ... Nt.Therefore Theorem 1.1 shows that Cρhas eventually polynomial enumeration, and so Cdoes as well.3 The Fibonacci dichotomy The Fibonacci dichotomy for permutation classes, first proved by Kaiser and...

... Furthermore, the cycle structure is obviouslypreserved. The seven patterns involved the above theorem share the property that they areequidistributed with the non-cyclic occurrences. Let Π(p) denote the ... unique pair of N and S steps. (The rightmostN step to the left of the F step ending at the F step’s level is paired with the leftmost Sstep to the right of the F starting at the F step’s level.) ... between cycles, 2–14 and 3–14, and 2 occurrences of 2–13 within cycles, 7–68 and 5–36.Let Φi,j(n) denote the number of permutations of length n, with i cyclic occurrencesof 2–13 and j cycles. Theorem...

... circuits and G is not ordinary. ✷Theorem 4.3 If neither G[X] nor G[X] is bipartite, then G is not ordinary.Proof: By hypothesis, b(G) + p(G) ≥ 2, and so the result follows from Theorem 2.3. ✷ The ... 1-factors.)In the next section we derive the existence or otherwise of the set S described earlierby relating the dimensions of the spaces L(G), C(G), E(G) and A(G).2 An argument using vector spaces Every ... brick. Then the unique brick in the decomposition of Ginto bricks and braces has the Petersen graph as its underlying simple subgraph if and only if there exists a complicating set of edges.the...