D. Stanton, D. White, Constructive Combinatorics, Springer Verlag 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One can see that the lattice of subsets of size k from the set of size n is a special case of Youngs s lattice when all i s are equal. Therefore, the number of integer partitions whose Ferrers diagrams t in a box of size k by n k is equal to (providing an alternate proof of Theorem 3. 2 in [SW86]) Let q(N; k; m) denote the number of partitions of N whose Ferrer s diagram t in a box of size k by m. By summing up the sizes of all level sets, we get = P k(n k) l=0 q(l; k; n k) Since the poset that generates corresponding Young s lattice is symmetric with respect to k and m, we get ....

D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.

.... 2. 2. Posets We will need some standard definitions and elementary results from the the theory of partially ordered sets (POSETS) All that we will need will be presented here, but the reader who wishes to know more about posets is referred to chapter 2 of Stanton and White s excellent textbook [12], and to Greene and Kleitman s well written survey article[4] A poset (P , is a set P with an order relation which has the following properties: i) a a for all a eP , ii) ab and ba implies a=b, and (iii) ab and bc implies ac. If a b and a b , we write a b . We say that b covers a if a ....

Stanton, Dennis, and White, Dennis, "Constructive Combinatorics", Springer-Verlag, New York, 1986.

....because the corresponding weights occur equally often with a positive and a negative sign. This is done by pairing up the bad clow sequences such that clow sequences which are matched have opposite signs. This proof technique is typical of many combinatorial proof for theorems in linear algebra [20, 18, 23, 25]. The following lemma is stated in Valiant [24] its proof is given in full detail in [14, 15] Lemma 1. Let C be a claw sequence that contains repeated elements. Then we can find a claw sequence with the same weight and opposite sign. Moreover, this mapping between C and is an involution, i.e. ....

Dennis Stanton, Dennis White, Constructive combinatorics. Springer-Verlag, New York 1986.

....function, the diameter of the corresponding digraph is tested and the best digraph is recorded. So far we have used this approach for permutations of eight elements. For the practical implementation we used the algorithms for ranking and unranking permutations described by Stanton and White in [22]. These algorithms enables us to store the graphs in a very compact way and to access any permutaion quickly. The resulting program is small and elTicient. 5 Conclusion The following tables sumarize the results obtained. The small size values of Table 1 are the best best vertex symmetric ....

D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, New York, 1986.

....because the corresponding weights occur equally often with a positive and a negative sign. This is done by pairing up the bad claw sequences such that claw sequences which are matched have opposite signs. This proof technique is typical of many cambinatarial proof for theorems in linear algebra [20, 18, 23, 25]. The following lemma is stated in Valiant [24] its proof is given in full detail in [14, 15] Lemma 1. Let C be a claw sequence that contains repeated elements. Then we can find a claw sequence C with the same weight and opposite sign. Moreover, this mapping between C and is an involution, ....

Dennis Stanton, Dennis White, Constructive combinatorics. Springer-Verlag, New York 1986.

....set K max = 6. In general, the value of K max depends on the application (see Section 5.5) Producing all subsets of a given image containing n objects is equivalent to generating all combinations of n elements taking them k at a time. Algorithms for producing combinations can be found in [107, 108]. Such algorithms have linear time complexity with respect to the number of combinations produced. For each value of k, i n k j = n (n;k) Deltak image subsets are produced. Taking all subsets for all values of k, results in a total of P min (n#Kmax ) k=2 i n k j subsets. For ....

....image subset p is mapped to a unique address in an address space of size k by computing the rank (order) of r with respect to a listing of permutations. For example, the rank corresponding to the ordered image subset (4 2 5 3) is 12. The ranking algorithm we used is that by Johnson and Trotter [108]. It has been assumed that no two objects have the same center of mass. The ordering of image subsets containing concentric objects (this may happen when one object contains another) is ambiguous and may result in different representations of subsets with similar spatial relationships (see also ....

Dennis Stanton and Dennis White. Constructive Combinatorics, chapter 1, pages 1--25. Springer-Verlag, 1986.

....have equal ranks. Thus, string s may take k different values. An ordered image subset p is mapped to a unique address I r#s k in an address space of size D r#s k = k by computing the rank (order) of r in a listing of permutations. The ranking algorithm we used is that by Johnson and Trotter [21]. For example, the ordered image subset p = 0125) derived from the image of Figure 1 has s = 0123) and I r#s k = 0, while D r#s 4 = 24. When the second ordering criterion is used, the spatial relationships between the objects contained in an ordered image subset are completely characterized ....

Dennis Stanton and Dennis White. Constructive Combinatorics, chapter 1. SpringerVerlag, 1986.

....image subset p is mapped to a unique address in an address space of size k by computing the rank (order) of r with respect to a listing of permutations. For example, the rank corresponding to the ordered image subset (4 2 5 3) is 12. The ranking algorithm we used is that by Johnson and Trotter [19]. It has been assumed that no two objects have the same center of mass. The ordering of image subsets containing concentric objects (this may happen when one object contains another) is ambiguous and may result in different representations of subsets with similar spatial relationships. In such ....

Dennis Stanton and Dennis White. Constructive Combinatorics, chapter 1, pages 1--25. Springer-Verlag, 1986.

....until (z) z or OE(y) y if (z) z then fl(x) z else fl(x) y else fx is not a fixed point of OE or g end. 1 This Lemma is extremely important because it not only establishes the existence of the involution fl but actually shows how to construct it. 1 Pseudo code quoted from [11], pages 141 142 27 Lemma 2 (The Bread Lemma) Given two sign reversing involutions, OE : A Gamma B A Gamma B and : B Gamma C B Gamma C, there is a sign reversing involution on A Gamma C. Proof . Let GammaI B represent the negative identity map on B Gamma B, extended to be the ....

D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.

....( 1) k # b e k m ## m m n b e k # = b # i=0 i # c=0 m i # a=0 n e b i c a # d=0 ( 1) c d # b i ## i c ## e a ## a d # . 11 Proof: By replacing k by m n b e k in the sum on the left hand side of the above equation and using the fact that [S W] j # d=0 ( 1) d # a d # = 1) j # a 1 j # (Notice that this means that # 1 j # = 1) j ) the problem is reduced to showing that b # i=0 i # c=0 m i # a=0 ( 1) n e b i a # b i ## i c ## e a # # a 1 n e b i c a # = m n b e # k=0 ( 1) k # m n k ....

.... that this means that # 1 j # = 1) j ) the problem is reduced to showing that b # i=0 i # c=0 m i # a=0 ( 1) n e b i a # b i ## i c ## e a # # a 1 n e b i c a # = m n b e # k=0 ( 1) k # m n k m ## m k # (13) Using the classical summation of Vandermonde [S W] m # k=0 # m k ## n i k # = # m n i # in the summation on c on the left hand side of (13) we find that (13) reduces to b # i=0 m i # a=0 ( 1) n e b i a # b i ## e a ## a i 1 n e b i a # = m n b e # k=0 ( 1) k # m n k m ## m k # ....

D. Stanton and D. White, "Constructive combinatorics," SpringerVerlag, New York, 1986.

....BT l , k l 1 = n on the sets [0; k] and [k 1; n] Let as define n (T ) as the binary tree whose left and right branches are equal to k (T 1 ) and l (T 2 ) correspondingly. See example on Figure 6.2. It is well known that the number of binary trees is equal to the Catalan number (e.g. see [SW]) y 6 0 1 2 3 4 5 6 Figure 6.2. Bijection between standard and binary trees. Now prove Theorem 6.3. Proof of Theorem 6.3. Recall that ffl 0 ; ffl 1 ; ffl n is the standard basis in Z n 1 ; and e ij = ffl i Gamma ffl j . Let e Pn ae Z n 1 Omega R denote the cone with vertex at ....

....8.6. Let I = f0; 2; 4; 2kg and J = f1; 3; 5; 2k 1g then D IJ is equal to the Catalan number C k . Proof. Words w 0 = w 0 1 ; w 0 2 ; w 0 2k ) of type (k; k) which exceed the word w = 1; 0; 1; 0; 1; 0) are called Dyck words. It is well know (see e.g. [SW]) that the Catalan number C k is equal to the number of Dyck words. 9. Standard Triangulation of P IJ Let I; J ae [0; n] I J = be two subsets such that min(I[J) 2 I and max(I[J) 2 J (see Section 8) Recall that P IJ = Conv(0; e ij : i; j) 2 S IJ ) HYPERGEOMETRIC FUNCTIONS ASSOCIATED ....

D. Stanton, D. White, Constructive Combinatorics, Springer-Verlag, 1986.

....integer lattice. Let be a partition such that 1 n and 0 1 m. As usual, 0 is the conjugate of ; hence 0 1 is the number of boxes in the first column of the Ferrers diagram of . For a description of partitions of an integer, see elementary combinatorics texts such as Stanton and White [4]. We shall consider all paths from the bottom left corner to the upper right corner such that each step is either up one unit or to the right one unit and such that the path does not go inside the Ferrers diagram of (placed so that the upper left corners of the Ferrers diagram and the m by n ....

Stanton, D., and White, D. Constructive Combinatorics. (Springer-Verlag, New York, 1986). 14

....4. However, if we examine the structure of the acyclic orientations of a specific graph, we can often find another way of computing a(G) We demonstrate by determining sufficient conditions to prevent AO(K m;n ) from being hamiltonian. Let S(n; k) denote the Stirling numbers of the second kind [SW86]. S(n; k) counts the number of set partitions of an n set into k blocks. Note that for n 1, S(n; 1) 1 and S(n; 2) 2 n Gamma1 Gamma 1. The formula k S(n; k) counts the number of ordered set partitions, which are set partitions in which the order of the blocks is important. We denote an ....

D. Stanton and D. White, "Constructive Combinatorics," Springer-Verlag (1986).

....proof of Theorem (1.12) Dyson [D2] has given a combinatorial proof of Theorem (1.12) for the case k = 2. More recently, he has proved (1.9) combinatorially [D4] Question 2. Find a combinatorial proof of Theorem (1.14) The case k = 1 has a well known and easy combinatorial proof. See [S W], A7] Lewis and Santa Gadea have found numerous relations between N V (m; n) and N 2 (m; n) Unfortunately we have found no further non trivial relations among the N k (m; n) Acknowledgments The author would like to thank George Andrews for suggesting the problem of generalizing Dyson s rank ....

D. Stanton and D. White, "Constructive Combinatorics," Springer-Verlag, New York, 1986.

....other objects. About three quarters of these functions deal with graphs and graph algorithms, the remaining quarter are devoted to other topics. The contents of this package make it quite suitable for a class in combinatorics and a good companion to the texts of Roberts [4] Stanton and White [5], or Tucker [6] I. Combinatorica and Mathematica. Combinatorica can be run only as part of a Mathematica session since it is written using primitives from that algebra package. I will assume the reader is familiar with Mathematica, its programming techniques and user interface. More information ....

....the complete tripartite graph K 2;2;2 is produced with the command In[4] ShowLabeledGraph[ K[2,2,2] whose output is shown in Figure 2. You will note that the edges f1,5g and f2,6g have been obscured. FIGURE 2 ABOUT HERE One can rectify this situation by shaking the graph and calling In[5]: ShowLabeledGraph[ ShakeGraph[ K[2,2,2] 0.2] which perturbs the vertices at random to eliminate collinearities (Figure 3) 5 FIGURE 3 ABOUT HERE It is a pity that the shaken version is not the default. Skiena provides a number of procedures for extracting graphical invariants. A single ....

Dennis Stanton and Dennis White, Constructive Combinatorics, SpringerVerlag, New York, NY, 1986.

....to be cyclic, or to go from the lexicographically minimum to maximum elements, properties not possessed by the Gray codes of Ehrlich and Knuth. 1 Introduction For a given integer n 0, a set partition is a decomposition of f1; ng as a disjoint union of nonempty subsets called blocks ([8], p. 18) The set of all partitions of f1; ng is denoted S(n) S(4) is listed in Figure 1(a) The restricted growth functions (RG functions) of length n, denoted R(n) are those strings a 1 : a n of non negative integers satisfying a 1 = 0 and a i 1 maxfa 1 ; a i Gamma1 g ....

....p. 18) The set of all partitions of f1; ng is denoted S(n) S(4) is listed in Figure 1(a) The restricted growth functions (RG functions) of length n, denoted R(n) are those strings a 1 : a n of non negative integers satisfying a 1 = 0 and a i 1 maxfa 1 ; a i Gamma1 g ([8], p. 18) With each 2 S(n) associate a string a 1 Delta Delta Delta a n as follows. Order the blocks of according to their smallest element, for example, the blocks of = f f9g, f1; 2; 7g, f4; 10; 11g, f3; 5; 6; 8g g would be ordered f1; 2; 7g, f3; 5; 6; 8g, f4; 10; 11g, f9g. Label the ....

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D. Stanton and D. White, Constructive Combinatorics, Springer-Verlag (1986).

....Let S(n) denote the set of all partitions of f1; ng. For example, S(4) is shown in Figure 9(a) The restricted growth functions (RG functions) of length n, denoted R(n) are those strings a 1 : a n of non negative integers satisfying a 1 = 0 and a i 1 maxfa 1 ; a i Gamma1 g [SW86]. There is a well known bijection between S(n) and R(n) For 2 S(n) order the blocks of according to their smallest element, for example, the blocks of = ff9g, f1; 2; 7g, f4; 10; 11g, f3; 5; 6; 8gg would be ordered f1; 2; 7g, f3; 5; 6; 8g, f4; 10; 11g, f9g. Label the blocks of in order by 0; ....

....there are strict Gray codes for R(n) and T (n; k) when the parity difference is 0. 21 7 Catalan Families In several families of combinatorial objects, the size is counted by the Catalan numbers, defined for n 0 by C n = 1 n 1 2n n : These include binary trees on n vertices [SW86], well formed sequences of 2n parentheses [SW86] and triangulations of a labeled convex polygon with n 2 vertices [STT88] Since bijections are known between most members of the Catalan family, a Gray code for one member of the family gives implicitly a listing scheme for every other member of ....

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D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.

....cost of exchanging the i th and the (i 1) st items is f(i 1) Gamma f(i) 3 Preliminaries It is convenient to regard a list L as a permutation of the numbers 1 to n. We give some useful definitions and results about permutations. General references for this section are [11, Ch. 5] and [15]. Let L 1 and L 2 be two different permutations. An inversion between L 1 and L 2 is a pair of items (x; y) such that x is in front of y in one list but behind y in the other. We write inv(L 1 ; L 2 ) for the set of inversions between L 1 and L 2 . The inversion table of a permutation L is a ....

.... table uniquely defines a permutation L and can be used to encode L by an integer n(L) 2 [0: n Gamma 1) by letting n(L) P n j=2 a j (j Gamma 1) If two adjacent items (x; y) are exchanged in L, giving (y; x) 2 Our definitions of inversion table and n(L) differ slightly from those in [11, 15] but are more convenient to program. in permutation L 0 , then n(L 0 ) can be derived from n(L) by adding (y Gamma 1) if x y, or subtracting (x Gamma 1) otherwise. The minimum number of exchanges of adjacent elements required to sort L is equal to jinv(L; h1; 2; ni)j. This ....

D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, New York, 1986.

....recently, Minoux [Min97] showed an extension of the matrix tree theorem to semirings, again using counting arguments over arborescences in graphs. For beautiful surveys of some of these results, see Zeilberger s paper [Zei85] and chapter 4 of Stanton and White s book on Constructive Combinatorics [SW86]. Zeilberger ends with a host of exercises in proving many more matrix identities combinatorially. Thus, using combinatorial interpretations and arguments to prove matrix identities has been around for a while. To our knowledge, however, a similar application of combinatorial ideas to interpret, ....

D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.

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D. Stanton, D. White, Constructive Combinatorics, Springer Verlag 1986.

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D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.

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D. Stanton and S. White. Constructive Combinatorics. Springer-Verlag, 1986.

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D. Stanton and D. White, Constructive Combinatorics, Springer-Verlag,

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Dennis Stanton and Dennis White. Constructive Combinatorics. Springer-Verlag, New York, 1986.

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D. W. Stanton and D. E. White, "Constructive combinatorics," SpringerVerlag, New York, 1986. -- 68 --

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