Douglas B. West. Generating linear extensions by adjacent transpositions. J. Comb. Theory Ser. B, 58:58-64, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... only by a single edge [HH72, Cum66] 5) listing all partitions of an integer n so that in successive partitions, one part has increased by one and one part has decreased by one [Sav89] 6) listing the linear extensions of certain posets so that successive elements differ only by a transposition [Rus92, PR91, Sta92, Wes93], and (7) listing the elements of a Coxeter group so that successive elements differ by a reflection [CSW89] Gray codes have found applications in such diverse areas as circuit testing [RC81] signal encoding [Lud81] ordering of documents on shelves [Los92] data compression [Ric86] statistics ....

....so far involve cutting and linking together listings for various subposets in rather intricate ways. In many cases where it is known how to list linear extensions by transpositions, it is also possible to require adjacent transpositions, although possibly with a more complicated construction [PR91, RS93, Sta92, Wes93]. It has been shown that if the linear extensions of a poset Q, with jQj even, can be listed by adjacent transpositions, then so can the linear extensions of QjP , for any poset P [Sta92] where QjP represents the union of posets P and Q with the additional relations fp q j p 2 P; q 2 qg. ....

D. B. West. Generating linear extensions by adjacent transpositions. Journal of Combinatorial Theory Series B, 57:58--64, 1993.

.... Los, Ric] There are many examples of combinatorial families for which Gray codes are known, including permutations [Joh, Tro] combinations [BuWi, NiWi, Rus1] compositions [Kli] set partitions [Kay] integer partitions [Sav, RaSaWe] binary trees [RuPr, Luc, LuRoRu] and linear extensions [PrRu1, PrRu2, Rus2, Sta, Wes]. When an application requires an exhaustive examination of all objects in a combinatorial family, Gray codes can be used to speed up the task. With a Gray code scheme, it is often possible to list a family of N objects, each of size O(n) in time O(n N ) rather than time O(n N ) by listing ....

D. B. West, "Generating linear extensions by adjacent transpositions," Journal of Combinatorial Theory (B), 57(1993).

....of a poset such that two consecutive linear extensions di er only by a transposition. They also considered the case where only adjacent transpositions are allowed, which is equivalent to nding a Hamilton path in G(P ) Hamilton paths in G(P ) were also investigated by Stachowiak [16] and West [18]. Bj orner and Wachs are the rst to consider structural properties of G(P ) apart from Hamiltonians. In their paper [1] they prove the fundamental theorem that the lattices of all extensions of a poset P is dually isomorphic to the lattice of all convex subsets (vertex sets of convex subgraphs, ....

D. B. West, Generating linear extensions by adjacent transpositions, J. Comb. Theory Ser. B, 58 (1993), pp. 58-64.

.... [5, 16] 2) generating bit strings by changing one bit [4, 3] 3) generating subsets by changing one element [1, 8, 12] 4) generating binary trees by rotations [7] 5) generating Coxeter group elements by reflection [2] and (6) generating linear extensions of certain posets by transpositions [9, 10, 13, 15, 17]. Such enumeration schemes are called minimal change algorithms or combinatorial Gray codes, in honor of the reflected binary code of Gray for solving problem (2) above. These schemes may permit efficient generation of combinatorial families of exponential size. They may list the elements of a ....

D. B. West, "Generating linear extensions by adjacent transpositions," Journal of Combinatorial Theory (B), 57(1993). 35

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Douglas B. West. Generating linear extensions by adjacent transpositions. J. Comb. Theory Ser. B, 58:58-64, 1993.

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