TAILIEUCHUNG - Báo cáo toán học: "Meet and Join within the Lattice of Set Partitions"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Meet and Join within the Lattice of Set Partitions. | Meet and Join within the Lattice of Set Partitions E. Rodney Canfield Department of Computer Science The University of Georgia Athens GA 30602 USA erca@ Submitted January 21 2001 Accepted March 1 2001. MR Subject Classifications 05A18 05A15 05A16 06C10 Abstract We build on work of Boris Pittel 5 concerning the number of i-tuples of partitions whose meet join is the minimal maximal element in the lattice of set partitions. 1 Introduction Recall that a partition of the set n 1 2 . n is a collection of nonempty pairwise disjoint subsets of n whose union is n . The subsets are called blocks. One partition n1 is said to refine another n2 denoted n1 n2 provided every block of n1 is contained in some block of n2. The refinement relation is a partial ordering of the set nn of all partitions of n . Given two partitions n1 and n2 their meet n1 A n2 respectively join n1 V n2 is the largest respectively smallest partition which refines respectively is refined by both n1 and n2. The meet has as blocks all nonempty intersections of a block from n1 with a block from n2. The blocks of the join are the smallest subsets which are exactly a union of blocks from both n1 and n2. Under these operations the poset nn is a lattice. Recently Pittel has considered the number Mfi of f-tuples of partitions whose meet is the minimal partition 1 2 . n and J the number of f-tuples whose join is the maximal partition 1 2 . n . We shall prove Theorem 1 Let Mt x and Jt x be the exponential generating functions for the sequences Mffi and JÍ . Then xn Mt ex - 1 YJB tn exp Jt x - 1 . where Bn is the n-th Bell number the total number of partitions of the set n . THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R15 1 Remark. What about n 0 and or t 0 The lattice n0 has exactly one element and thus is isomorphic to n1. Generally one takes the empty meet to be the maximal element and the empty join to be the minimal element. Thus there is some logical justification to define m0Í J 1 for all t M Jo

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