doc/combinatorics.tex

   1 \documentclass[11pt]{kasper}
   2 \usepackage{relsize}
   3
   4 \begin{document}
   5 \title{{\tt combinatorics.hh} documentation}
   6 \author{Kasper Peeters}
   7 \address{1}{{\it MPI/AEI f\"ur Gravitationsphysik, Am M\"uhlenberg 1, 14476
   8 Potsdam, Germany}}
   9 \email{k.peeters@damtp.cam.ac.uk}
  10 \maketitle
  11 \begin{abstract}
  12 The {\tt combinatorics.hh} class is bla bla
  13 \end{abstract}
  14
  15 \begin{sectionunit}
  16 \title{Reference guide}
  17 \maketitle
  18 The {\tt combinatorics.hh} class library contains two classes for the
  19 generation of combinations
  20 \begin{sectionunit}
  21 \title{The {\tt combinations} class}
  22 \maketitle
  23 The {\tt combinations} class takes a vector of elements, and generates
  24 combinations of a subset of these elements that satisfy several
  25 requirements.
  26 \begin{description}
  27 \item[{\tt sublengths}]~\\
  28 Stores the lengths of the object ranges in which objects can only
  29 appear in increasing order. The usual concepts of ``combinations and
  30 permutations'' translate to the following content of the {\tt
  31 sublengths} vector:
  32 \begin{equation}
  33 \begin{aligned}
  34 \text{combinations:} &\quad \text{one element, $v$=total length}\, ,\\
  35 \text{permutations:} &\quad \text{$n$ elements, all equal to $1$}.
  36 \end{aligned}
  37 \end{equation}
  38 However, many more general situations are possible with {\tt
  39   combinatorics.hh}. See the examples below.
  40
  41 \item[{\tt input\_asym}] ~\\
  42 Stores sets of objects which have to be ``implicitly
  43 permuted''. That is, permutations are done only modulo the
  44 permutations of the objects in these sets. An example may help:
  45 \begin{equation}
  46 \begin{aligned}
  47 \{a,b,c\}           &\rightarrow \frac{1}{6}(a,b,c + a,c,b + b,a,c + c,a,b + b,c,a + c,b,a)\\
  48 \{a,b,c\}_{\{0,1\}} &\rightarrow \frac{1}{3}(a,b,c + a,c,b + c,a,b)\\
  49 \{a,b,c\}_{\{0,2\}} &\rightarrow \frac{1}{3}(a,b,c + a,c,b + b,a,c)\\
  50 \{a,b,c\}^{\{2,1\}} &\rightarrow \frac{1}{3}(a,b,c + a,c,b + c,a,b)
  51 \end{aligned}
  52 \end{equation}
  53 Note that this does not just remove terms, it also corrects the
  54 multiplicative factor.
  55 \end{description}
  56 \end{sectionunit}
  57 \begin{sectionunit}
  58 \title{The {\tt symmetriser} class}
  59 \maketitle
  60 The {\tt symmetriser} class takes a vector of elements, and applies
  61 permutations or combinations to it. The difference with {\tt
  62   combinations} is that {\tt symmetriser} can be used to apply several
  63 permutations successively.
  64
  65 Here is an example,
  66 \begin{screen}
  67 symmetriser<string> sm;
  68 sm.original.push_back("r1");
  69 sm.original.push_back("r2");
  70 sm.original.push_back("r3");
  71 sm.original.push_back("r4");
  72 sm.original.push_back("r5");
  73
  74 sm.block_length=1;
  75 sm.permutation_sign=-1;
  76 sm.permute_blocks.push_back(1);
  77 sm.permute_blocks.push_back(2);
  78 sm.apply_symmetry();
  79 \end{screen}
  80 This permutes the {\tt r2} and {\tt r3} items, with a minus sign under
  81 exchange, leaving the other elements unchanged.
  82
  83 \begin{description}
  84 \item[set storage]
  85 \item[result storage]
  86 \item[permuting by position]
  87 \item[permuting by value]
  88 \item[weights]
  89 \item[input symmetries] Just as with the {\tt combinations} class,
  90 the {\tt symmetriser} class can be told to apply symmetries only
  91 modulo given symmetries. Take care when permuting objects by value:
  92 the \verb|input\_asym| data still refer to object locations, not
  93 object values (this may be a tricky bug to find when you are permuting
  94 integers). The object locations are always relative to the set of
  95 permuted values, not with respect to the complete set of original values.
  96 \end{description}
  97
  98
  99 \end{sectionunit}
 100 \end{sectionunit}
 101
 102 \begin{sectionunit}
 103 \title{Examples}
 104 \maketitle
 105
 106 \end{sectionunit}
 107 \end{document}