Let's Expand Rota's Twelvefold Way For Counting Partitions!
Let's Expand Rota's Twelvefold Way
For Counting Partitions!
Abstract
Rota's Twelvefold Way gave formulas for the numbers of partitions which could be formed in twelve scenarios. This proposed AMM article expands Rota's 4 x 3 table. The resulting 6 x 5 table considers a broader collection of splitting- distributing- grouping- arranging scenarios, each of which can be visualized with the distribution of m items into certain kinds of bins. The additional counts or scenarios include: the Bell numbers B(m), the partition numbers p(m), arrangements of m books on b shelves, standings of m teams in a league, arrangements of m books into b scattered stacks, and pairings of 2m items. Teaching remarks are included. The two additional rows (due to K. Bogart) consider ordering the items within the bins. One additional column distributes the items into an unspecified number of bins, each receiving at least one item. The other (due to T. Brylawski) distributes the items into bins such that the number of bins containing a given number of items is specified. The quotient and summation relationships amongst the thirty counts are stated. A closely related table formed by the same six rows and seven certain columns is used to complete and to organize a 6 x 7 family of counting sequences in the On-Line Encyclopedia of Integer Sequences.
The Article
Submitted to Monthly January 5, 2006 edition. 26 pages.
PDF version 1.4 (for Adobe Acrobat Reader 5.0 or later).
The characters are blackest when printed on 8.5" x 11" paper.
The layout is the nicest when printed two-sided (back-to-back).
165 kb non-compressed file. Click here to
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the article's pdf file now.
June 16, 2006 edition.
arXiv number:
math.CO/0606404.
26 pages. PDF version 1.4 (for Adobe Acrobat Reader 5.0 or later).
Class Projects
These two projects are briefly described in Section 6 of the article.
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the 6 page Microsoft Word document containing these class projects.
Please note that the 4th page contains the answers to the problems posed in the\
first project!
Tables 1 and 2
If you want to modify Table 1 or Table 2 for your own purposes:
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the Microsoft Word document containing these two tables.
Note: Vertical table lines are formed from bold pipe characters. Horizontal table lines are formed by selecting all of the relevant tab characters and choosing 'underline'. The easiest way to modify these tables is as follows: Start with an empty document. Copy a generic line from Table 1. Use the Format/Tabs menu to reposition the column dividers, etc within this line. Duplicate the modified row as many times as needed. Keep the tabs in your rows identical for as long as possible. When you need to add some the tab stops in the same way for more than one row (as for the summation limits in Table 2), use the keystroke version of the "again" command (which is officially referred to as the EditRedoOrRepeat command under Tools/Customize/Keyboard) after moving the cursor to the next line to be similarly revised. This will repeat all of the revisions made during your immediately preceding visit to the Format/Tabs dialog box. If you mess things up, it will not take long to start out fresh with another empty document.