By Flajolet Ph., Sedgewick R.
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Additional resources for Analytic combinatorics - symbolic combinatorics
This is reflected by the use of a sequence construction (for ❭ ) against a multiset construction (for ❪ ). In this perspective, it proves convenient to regard ò as obtained by the empty sequence of summands (✟ ◆ ï➷ò ), and we shall do so from now on. ❛ õ First, let ✦ denote the combinatorial class of all integers at least 1 (the ❫ ï❵Û❴ ð ✆ size ✆ ✺▲✺❀✺ of each integer be its value. Then, the OGF of ❫ is summands), and let the ú õ ❝ ❝ ❝ ï ❜ û➵ï❡❞ (13) õ ð❤❣ ❝ ✆ ❢ ❑ ø since ❜ ï➜ð for î ❚ ð , corresponding to the fact that there is exactly one object in ❫ for each size î ❚ ð .
Graphically, compositions may be seen as as “ragged-landscapes” (represent the summands vertically) or equivalently as alignments of balls with dividing lines, the “ballsand-bars” model; in contrast, partitions appear as “staircases” also known as Ferrers diagrams [28, p. 100]; see Figure 5. We let ❭ and ❪ denote the class of of all compositions and all partitions. Since a set can always be presented in sorted order, the difference between compositions and partitions lies in the fact that the order of summands does or does not matter.
2. Integer related constructions. Finally, we say a few words about the two constructions of cycle and powerset that haven’t been yet applied to ❫ . First, the class ❩ ï ● ❴➲❜ ❛ comprises cyclic compositions, that is, compositions defined up to circular shift; so, for instance ö ö✐ ❩ ✈ð♠ö ö ✈ , ✐ ö✈ð♠ö ö ✈ ö , etc, are identified. Alternatively, we may view elements composed ú of as “wheels” ú ú of ú circular arrangements of segments (taken up to circular symmetry). 6 ▼ ❣ ❖▲❖❀❖ ❣❳❏✠❑ , so that the OGF is but with the strict inequalities ❏✠❤ õ ✐ ❝ û❆ï ➉ ð ö ❝ û ❢ ❑ ✺ ø ø 28 I.