Symbolic method (combinatorics): Difference between revisions

Revision as of 23:19, 31 July 2019

Revision as of 03:13, 1 August 2019

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In the following we will simplify our notation a bit and write e.g.

:<math> E_2 + E_3 \text{ and } C_1 + C_2 + C_3 + \cdots.</math>

for the classes mentioned above.

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:<math>

F(z) = \sum_{n\ge 1} Z(C_n)(f(z), f(z^2), \ldots, f(z^n)) =

\sum_{n\ge 1} \frac{1}{n} \sum_{d|\mid n} \varphi(d) f(z^d)^{n/d}</math>

or

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Evaluating <math>M(f(z), 1)</math> we obtain

:<math> F(z) = \exp \left( \sum_{l\ell\ge 1} \frac{f(z^l\ell)}{l\ell} \right).</math>

For the labelled case we have

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In the labelled case we denote the operator by {{math|SET}}, and in the unlabelled case, by {{math|MSET}}. This is because in the labeled case there are no multisets (the labels distinguish the constituents of a compound combinatorial class) whereas in the unlabeled case there are multisets and sets, with the latter being given by

:<math> F(z) = \exp \left( \sum_{l\ell\ge 1} (-1)^{l\ell-1} \frac{f(z^l\ell)}{l} \ell \right).</math>

==Procedure==

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A class of combinatorial structures is said to be ''constructible'' or ''specifiable'' when it admits a specification.

For example, the set of trees whose leaves's depth is even (respectively, odd) can be defined using the specification with two classes <math>\mathcal A_\text{even}</math> and <math>\mathcal A_\text{odd}</math>. Those classes should satisfy the equation <math>\mathcal A_\text{odd}=\mathcal Z\times \mathrmoperatorname{Seq}_{\ge1}\mathcal A_\text{even}</math> and <math>\mathcal A_\text{even} = \mathcal Z\times \mathrmoperatorname{Seq}\mathcal A_\text{odd}</math>.

==Labelled structures==

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===Boxed product===

In labelled structures, the min-boxed product <math>\mathcal{A}_{\min} = \mathcal{B}^{\square}\star \mathcal{C}</math> is a variation of the original product which requires the element of <math>\mathcal{B}</math> in the product with the minimal label. Similarly, we can also define a max-boxed product, denoted by <math>\mathcal{A}_{\max} = \mathcal{B}^{\blacksquare} \star \mathcal{C}</math>, by the same manner. Then we have,

:<math>A_{\min}(z)=A_{\max}(z)=\int^z_0 B'(t)C(t)\,dt.</math>

or equivalently,

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:<math> \operatorname{SET}(\operatorname{CYC}(\mathcal{Z}))</math>

is used to study unsigned [[Stirling numbers of the first kind]], and in the derivation of the [[Random permutation statistics|statistics of random permutations]]. A detailed examination of the [[exponential generating function]]s associated to Stirling numbers within symbolic combinatorics may be found on the page on [[Stirling numbers and exponential generating functions in symbolic combinatorics]].

the [[Random permutation statistics|statistics of random permutations]].

A detailed examination of the [[exponential generating function]]s associated to Stirling numbers within symbolic combinatorics may be found on the page on [[Stirling numbers and exponential generating functions in symbolic combinatorics]].

==See also==