# funs.pijPij: The functions for probability of selecting a number of points... In nnspat: Nearest Neighbor Methods for Spatial Patterns

## Description

The ancillary probability functions used in computation of the variance-covariance matrices of various NN spatial tests such as NNCT tests and tests based on other contingency tables. These functions can be classified as `pij` and `Pij` type functions. The `pij` functions are for individual probabilities and the corresponding `Pij` functions are the summed `pij` values. For example p_{iijk} is the probability of any 4 points with 2 from class i, and others are from classes j and k. These probabilities are for data from RL or CSR.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43``` ```p11(k, n) P11(nvec) p12(k, l, n) P12(nvec) p111(k, n) P111(nvec) p1111(k, n) P1111(nvec) p112(k, l, n) P112(nvec) p122(k, l, n) p123(k, l, m, n) P123(nvec) p1234(k, l, m, p, n) P1234(nvec) p1223(k, l, m, n) p1123(k, l, m, n) P1123(nvec) p1122(k, l, n) P1122(nvec) p1112(k, l, n) P1112(nvec) ```

## Arguments

 `k, l, m, p` Positive integers, usually representing the class sizes, used in `pij` type functions only. Number of these arguments required depends on the number of distinct indices of p, e.g. p_{ij} requires `k,l,n` and p_{iijk} requires `k,l,m,n` as input. `n` A positive integer representing the size of the data set (i.e., number of observations in the data set). `nvec` A `vector` ofpositive integers representing the sizes of classes in the data set, used in `Pij` type functions only.

## Value

Probability values for the selected points being from the indicated classes.

`pk`