Báo cáo toán học: "Escher''''''''s Combinatorial Patterns" pot

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Ngày tải lên: 05/08/2014, 09:46

... ,q |Q| }. 4.Ateverystageofthisconstructionthenumberofpolytopesgrowsbythenumberof clonings;ifatthatstageonehadRpolytopesthepointchosenwillbeinatleasthR ofthemandtherewillbeatmost(1−h)Rclonings.Thenumberofpolytopesinthat stagewillgrowtoatmost(2−h)R. 5.ThusthefinalnumberofpolytopeswillbeatmostN(2−h) |Q| . 6.Wewillusethefactthatthenumberofclonedescendantsofanyoneofouroriginal polytopescannotexceed...

Ngày tải lên: 07/08/2014, 06:22

... idea. In all these theorems, the underlying generating functions were either modular forms or simple linear combinations thereof. The point of this paper is to show that Ramanujan’s original method ... to an in nite number of congruence theorems including many non-modular functions defined by q-series. Our main result is: Theorem 1. Suppose p is a prime > 3,and0<a<pand b are in...

Ngày tải lên: 07/08/2014, 06:22

... played the game with his father.) Two pieces of cut potato can serve as the medium on which to carve the motif and its reflected image, and then these potato stamps are inked and used to produce a ... translations of the translation block. These symmetries add a geometric layer of complexity to the combinatorial scheme. Escher's Case (1) We first consider Escher's case (1), in whi...

Ngày tải lên: 07/08/2014, 06:22

Ngày tải lên: 07/08/2014, 06:22

... which, miraculously, Franklin’s argument proves at once. The author wishes to thank George Andrews and Don Zagier for supplying him with copies of [3] and [8], and also an anonymous referee for ... [1] and presented in more detail in [8]. Zagier [8] deduces (2) from (8), essentially by carefully differentiating with respect to x and setting x =1. Franklin’s argument proves an i...

Ngày tải lên: 07/08/2014, 06:22

... similarly. Question 6.4. For what conditions on a poset P are the statistics des and dmc equidis- tributed on the set of linear extensions of P? One might apply another variation of the method ... following equidistribution result involving the statistics inv (inversions) and maj (major index). These two statistics belong to the class of Mahonian statistics. (See [8] for further infor- ma...

Ngày tải lên: 07/08/2014, 06:22

... =1,then(S) 1 ⊆ ∗ n . Hence this special case of Zibin’s Conjecture is Toida’s Conjecture, so that Zibin’s Conjecture implies Toida’s Conjecture. Corollary 2.9 Toida’s Conjecture implies Zibin’s Conjecture. Proof. ... δ(B)=B, δ is also an isomorphism of  X( n ,S−∪{(S) r : r is a multiple of m})and  X  ( n ,T−∪{(T ) r : r is a multiple of m}), the d-th layer of  X( n ,S −...

Ngày tải lên: 07/08/2014, 07:21

... 33111 322 24 → 331 322 2411 → 331 324 221 1 → 3431 322 211 → 3431 322 211 ∈ [4] 10 (1 12) . (7) the electronic journal of combinatorics 9 (2) (20 02) , #R3 15 Corollary 16 For any p ≥ 0 F v p,0 (x; 2) = e x p  j=0 x j j! . Proof. ... 121 }, {1 12, 121 }, {1 12, 122 }, {1 12, 21 1}, {1 12, 21 2}, {1 12, 22 1}, { 121 , 21 2}. Theorem 5 The pairs {111, 1 12} and {111, 121 } are Wilf e...

Ngày tải lên: 07/08/2014, 07:21

... work of Mansour [14], who computed generating functions for the number of n-permutations that avoid two distinct patterns of length three and contain exactly one copy of another pattern of any length. It ... avoid two distinct patterns of length three and any additional set of patterns. We begin by showing how to enumerate these permuta- tions using generating tre...

Ngày tải lên: 07/08/2014, 07:21

... Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), J. Comput. Appl. Math. 49:1–3 (1993), 243–249. [6] A. L. Schmidt, “Legendre transforms and Ap´ery’s sequences”, J. Austral. Math. ... On a combinatorial problem of Asmus Schmidt W. Zudilin ∗ Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2, Moscow 119992 RUSSIA URL:...

Ngày tải lên: 07/08/2014, 08:20

... vertex. 2. The initial vertex of the next edge is v(j, ∓k). 3. Determine the direction of the next edge by the sign of the entry ∓k in list j and the direction of the preceding edge according to the ... combining the information about the direction of the first edge with the corresponding crossing yields a rule for the direction of the second edge. If...

Ngày tải lên: 07/08/2014, 08:20

... symbol of σ.Sincen +1>m, we can select a random subsequence of σ of length m by first throwing out one of the n + 1 symbols of σ at random and then randomly selecting a subsequence of length ... the following theorem of Stromquist. We give a proof which is somewhat shorter than the proof in [1], both for its own interest and because we will reuse the method of proof when stu...

Ngày tải lên: 07/08/2014, 08:22

... system. Generalizations of Theorems 1 and 2 are stated next without proof. The proofs for these are very similar to the ones used for the original theorems. Indeed, the only change to the original proofs is ... higher order, intersecting set partitions. 1 Introduction In this paper, we prove two Erd˝os-Ko-Rado type theorems for systems of uniform set partitions. They are stated a...

Ngày tải lên: 07/08/2014, 13:21

... #R63 29 Kocay’s lemma, Whitney’s theorem, and some polynomial invariant reconstruction problems Bhalchandra D. Thatte Allan Wilson Centre for Molecular Ecology and Evolution, and Institute of ... (2),  G F  and  G − u F  may be replaced by  G F  and  G − u F  , respectively. Tutte [14], [15] proved the reconstructibility of the characteristic polynomial and...

Ngày tải lên: 07/08/2014, 13:21