Generalized Frobenius partitions, k-cores, k-quotients, and cranks
نویسندگان
چکیده
منابع مشابه
k-Efficient partitions of graphs
A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...
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New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related...
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Alon, Bollobás, Krivelevich and Sudakov [1] proved that every graph with a large cut has a bipartition in which each vertex class contains correspondingly few edges. We prove an analogous result for partitions into k ≥ 3 classes; along the way we prove a result for biased bipartitions.
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We prove that almost surely the first non-empty (k+1)-core to arise during the random graph process will have a k-factor or will be k-factor-critical. Thus the threshold for the appearance of a k-regular subgraph is at most the threshold for the (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobas, Kim and Verstraete [3].
متن کاملON k-NONCROSSING PARTITIONS
In this paper we prove a duality between k-noncrossing partitions over [n] = {1, . . . , n} and k-noncrossing braids over [n − 1]. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to tangled-diagrams [6]. We give a combinatorial interpretation of the bijection in terms of the contraction of arcs of tangled-diagrams. Furthermore it induces by re...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1992
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-62-1-97-102