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27
A Survey of Combinatorial Gray Codes
 SIAM Review
, 1996
"... The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that ..."
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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems. ...
SignBalanced Posets
 J. COMBINATORIAL THEORY SER. A
, 2000
"... Let P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P . Call P signbalanced if the number of linear extensions of P of positive sign is the same as the number of linear extens ..."
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Let P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P . Call P signbalanced if the number of linear extensions of P of positive sign is the same as the number of linear extensions of P of negative sign. In this paper we determine when the posets in a particular class are signbalanced. When posets in this class are not signbalanced, we determine the difference between the number of positive linear extensions and the number of negative linear extensions. One special case of this class is the product of an mchain with an nchain, m and n both ? 1. In this case, we show P is signbalanced if and only if m = n mod 2.
Equidistribution and SignBalance on 321Avoiding Permutations
, 2004
"... Let Tn be the set of 321avoiding permutations of order n. Two properties of Tn are proved: (1) The last descent and last index minus one statistics are equidistributed over Tn, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for ..."
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Cited by 11 (2 self)
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Let Tn be the set of 321avoiding permutations of order n. Two properties of Tn are proved: (1) The last descent and last index minus one statistics are equidistributed over Tn, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The signandlastdescent enumerators for T2n and T2n+1 are essentially equal to the lastdescent enumerator for Tn. The proofs use a recursion formula for an appropriate multivariate generating function.
On Arrangements Of Roots For A Real Hyperbolic Polynomial And Its Derivatives
, 2001
"... . In this paper we count the number ] (0;k) n ; k n 1 of connected components in the space (0;k) n of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arra ..."
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Cited by 11 (6 self)
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. In this paper we count the number ] (0;k) n ; k n 1 of connected components in the space (0;k) n of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p(x) together with all its nonvanishing derivatives under the assumption that the roots of p(x) are real. Already the rst nontrivial case n = 4 shows that the obvious restrictions coming from the standard Rolle's theorem are insucient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials. x1.
A Gray Code for Necklaces of Fixed Density
 SIAM J. Discrete Math
, 1997
"... A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all nbit necklaces with d ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the las ..."
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Cited by 8 (0 self)
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A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all nbit necklaces with d ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the last and the first in the list, differ only by the transposition of two bits. The total time required is O(nN (n; d)), where N (n; d) denotes the number of nbit binary necklaces with d ones. This is the first algorithm for generating necklaces of fixed density which is known to achieve this time bound. 1 Introduction In a combinatorial family, a Gray code is an exhaustive listing of the objects in the family so that successive objects differ only in a small way [Wil]. The classic example is the binary reflected Gray code [Gra], which is a list of all nbit binary strings in which each string differs from its successor in exactly one bit. By applying the binary Gray code, a variety of problems...
Loopless generation of multiset permutations using a constant number of variables by prefix shifts
 In SODA ’09: The Twentieth Annual ACMSIAM Symposium on Discrete Algorithms
, 2009
"... This paper answers the following mathematical question: Can multiset permutations be ordered so that each permutation is a prefix shift of the previous permutation? Previously, the answer was known for the permutations of any set, and the permutations of any multiset whose corresponding set contains ..."
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This paper answers the following mathematical question: Can multiset permutations be ordered so that each permutation is a prefix shift of the previous permutation? Previously, the answer was known for the permutations of any set, and the permutations of any multiset whose corresponding set contains only two elements. This paper also answers the following algorithmic question: Can multiset permutations be generated by a loopless algorithm that uses sublinear additional storage? Previously, the best loopless algorithm used a linear amount of additional storage. The answers to these questions are both yes.
Discriminant sets of families of hyperbolic polynomials of degree 4 and 5
 Serdica Math. J
"... To the memory of my mother Abstract. A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible nondegenerate configurations between the roots of a degree 5 strictly hyp ..."
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To the memory of my mother Abstract. A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible nondegenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study the hyperbolicity domain of the family P (x; a, b, c) = x5 − x3 + ax2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P (i), P (j)) = 0, 0 ≤ i < j ≤ 4. 1. Introduction. 1.1. Statement of the problem. Definition 1. A real polynomial of degree n of one real variable is called
Shift Gray Codes
, 2009
"... Combinatorial objects can be represented by strings, such as 21534 for the permutation (1 2) (3 5 4), or 110100 for the binary tree corresponding to the balanced parentheses (()()). Given a string s = s1s2⋯sn, the rightshift operation ��→ shift(s, i, j) replaces the substring sisi+1⋯sj by si+1⋯sjsi ..."
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Cited by 7 (4 self)
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Combinatorial objects can be represented by strings, such as 21534 for the permutation (1 2) (3 5 4), or 110100 for the binary tree corresponding to the balanced parentheses (()()). Given a string s = s1s2⋯sn, the rightshift operation ��→ shift(s, i, j) replaces the substring sisi+1⋯sj by si+1⋯sjsi. In other words, si is rightshifted into position j by applying the permutation (j j −1 ⋯ i) to the indices of s. Rightshifts include prefixshifts (i = 1) and adjacenttranspositions (j = i + 1). A fixedcontent language is a set of strings that contain the same multiset of symbols. Given a fixedcontent language, a shift Gray code is a list of its strings where consecutive strings differ by a shift. This thesis asks if shift Gray codes exist for a variety of combinatorial objects. This abstract question leads to a number of practical answers. The first prefixshift Gray code for multiset permutations is discovered, and it provides the first algorithm for generating multiset permutations in O(1)time while
BINARY BUBBLE LANGUAGES AND COOLLEX ORDER
"... A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees ..."
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A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees, unit interval graphs, linearextensions of Bposets, binary necklaces and Lyndon words, and feasible solutions to knapsack problems. In colexicographic order, fixeddensity binary strings are ordered so that their suffixes of the form 10i occur (recursively) in the order i = max,max −1,...,min +1,min for some values of max and min. In coollex order the suffixes occur (recursively) in the order max −1,..., min+1, min, max. This small change has significant consequences. We prove that the strings in any bubble language appear in a Gray code order when listed in coollex order. This Gray code may be viewed from two different perspectives. On the one hand, successive binary strings differ by one or two transpositions, and on the other hand, they differ by a shift of some substring one position to the right. This article also provides the theoretical foundation for many efficient generation algorithms, as well as the first