C. W. Ko, F. Ruskey, Solution of some multi--dimensional lattice path parity di#erence recurrence relations, Discrete Math. 71 (1988), 47-56.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem is also open for posets whose Hasse diagram is a grid or tableau tilted ninety degrees [Rus] Calculating the parity difference itself can be difficult and Ruskey [Rus] has several examples of posets for which the parity difference is unknown. Some parity differences are calculated in [KR88]. Recently, Stachowiak has shown that computing the parity difference is #P complete [Sta] Even counting the number of linear extensions of a poset is an open problem for some specific posets, for example, the Boolean lattice [SK87] Brightwell and Winkler have recently shown that the problem of ....

C. W. Ko and F. Ruskey. Solution of some multi-dimensional lattice path parity difference recurrence relations. Discrete Mathematics, 71:47--56, 1988.

....form is ( Gamma1) n Gammak D(n Gamma 1; k Gamma 1) Lemma 2 For 1 k n D(n; k) 0 if n odd, k even, or n odd, k = 1 T (bn=2c; dk=2e) otherwise (6) Proof: An easy, if somewhat tedious, inductive proof using (5) can be given. An alternative proof is to use Theorem 2 of Ko and Ruskey [5], after noticing that T(n; k) is isomorphic to the set of ordered forests with k trees and 2n Gamma k nodes, where the number of vertices with (out)degree two is n Gamma k and the number of leaves (e.g. nodes of degree 0) is n. Yet another proof can be derived by using an involution similar to ....

C.W. Ko and F. Ruskey. Solution of some multi-dimensional lattice path parity difference recurrence relations. Discrete Math., to appear.

....Note that this is always an induced subgraph of the transposition graph of all permutations. The problem of generating all the linear extensions of a poset is now transformed into that of finding a Hamiltonian path in the transposition graph of the poset. It has been observed by Ko and Ruskey [12] and Lehmer [13] in slightly different contexts that the transposition graph is bipartite. In the present context it is easy to see that the transposition graph is bipartite, since a transposition reverses the parity of a permutation. If the difference of the number of vertices in the two partite ....

....be sufficient. In the fourth section we consider posets whose Hasse diagrams consist of disjoint chains. As noted above, the linear extensions of these posets correspond to multiset permutations. In Section 4. 1 the necessary condition for the existence of a Hamilton path given by Ko and Ruskey [12] is shown to also be sufficient. This is the most significant result of the paper. In Section 4.2 we restrict our attention to adjacent transpositions. Partial results on the existence of Hamiltonian paths and cycles are stated. The final section mentions some open problems. 2 General Partial ....

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C.W. Ko and F. Ruskey. Solution of some multi-dimensional lattice path parity difference recurrence relations. Discrete Math., 71:47--56, 1988.

....t ) d(k Gamma i; n 0 ; n t Gamma1 ) if t 0 odd P min(n t ;k) i=max(0;k Gamman n t ) Gamma1) i d(k Gamma i; n 0 ; n t Gamma1 ) if t 0 even This recurrence can be solved if all n i = 1. See, for example, Buck and Wiedemann [2] Eades, Hickey and Read [3] Ko and Ruskey [10]. Then d(k; 1) 0 if t and k are odd i b(t 1) 2c bk=2c j otherwise: By similar reasoning, it can also be solved when all n i = 1. d(k; 1) 0 if t and k are odd i b(t k) 2c bk=2c j otherwise We close by mentioning three open problems: ffl Determine exact conditions under which ....

C.W. Ko and F. Ruskey, Solution of Some Multi-dimensional Lattice Path Parity Difference Recurrence Relations, Discrete Mathematics, 71 (1988) 47-56.

....Note that this is always an induced subgraph of the transposition graph of all permutations. The problem of generating all the linear extensions of a poset is now transformed into that of finding a Hamiltonian path in the transposition graph of the poset. It has been observed by Ko and Ruskey [10] and Lehmer [11] in slightly different contexts that the transposition graph is bipartite. In the present context it is easy to see that the transposition graph is bipartite, since a transposition reverses the parity of a permutation. If the difference of the number of vertices in the two partite ....

....be sufficient. In the fourth section we consider posets whose Hasse diagrams consist of disjoint chains. As noted above, the linear extensions of these posets correspond to multiset permutations. In Section 4. 1 the necessary condition for the existence of a Hamilton path given by Ko and Ruskey [10] is shown to also be sufficient. This is the most significant result of the paper. In Section 4.2 we restrict our attention to adjacent transpositions. Partial results on the existence of Hamiltonian paths and cycles are stated. The final section mentions some open problems. 2 General Partial ....

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C.W. Ko and F. Ruskey. Solution of some multi-dimensional lattice path parity difference recurrence relations. Discrete Math., 71:47--56, 1988.

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C. W. Ko, F. Ruskey, Solution of some multi--dimensional lattice path parity di#erence recurrence relations, Discrete Math. 71 (1988), 47-56.

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