REMARKS ON THE INPUT ARGUMENTS.

Input checks are intentionally avoided, as lexunrank is supposed to be called many times, for sampling subsets. Thus, please ensure that: - k < n;

- N is an integer between 0 and bc(n,p)-1.

It is possible to enable checks, by changing an internal "if" statement to 1.

REMARKS ON THE OUTPUT ARGUMENTS.

As $n$ increases, 'calls' becomes much smaller than 'ncomb'. This means that lexunrank(n,k,N) is extremely convenient if you are interested in one or several, but not all, $k$-combinations at given generation order(s) N.

To generate all combinations in lexicographic order, it is more convenient using the FSDA function combsFS. The MATLAB function with the same purpose, nchoosek(1:4,3), is much less efficient.

ON THE LEXICOGRAPHIC ORDERING.

lexunrank(n,k,N) gives the $k$-combination of n elements of position N in the reverse co-lexicographic order of such combinations or, equivalently, of position bc(n,k)-N in the lexicographic order of the same combinations.

Note that, in this implementation of the lexicographic unrank, N ranges over the integers between 0 and bc(n,k)-1. For details see the "combinatorial number system" discussed by Knuth (2005), pp. 5-6.

To clarify with an example the meaning of the different orders, while the lexicographic order of the 2-combinations of 3 elements are:

\[ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 1 & 3 & 4 \\ 2 & 3 & 4 \end{array} \right) \] the co-lexicographic order of the same combinations are \[ \left( \begin{array}{ccc} 3 & 2 & 1 \\ 4 & 2 & 1 \\ 4 & 3 & 1 \\ 4 & 3 & 2 \end{array} \right) \] and the reverse co-lexicographic order of the original combinations are: \[ \left( \begin{array}{ccc} 4 & 3 & 2 \\ 4 & 3 & 1 \\ 4 & 2 & 1 \\ 3 & 2 & 1 \end{array} \right) \]

The reasons for choosing a co-lexicographic unrank is that right-to-left array filling is much faster and elegant. The reverse is due to a similar motivation.

ALGORITMIC DETAILS.

Given the totally ordered set $S=\{1,2,\ldots,n\}$, a $k$-combination is a subset $\{x_1, \ldots, x_k\}$ of $S$. Consider the $n$-lists of elements of the set $\{0,1\}$, i.e. the vertices of the hypercube $V_n$.

Each $k$-combination $\{x_1,\ldots,x_k\}$ can be associated to the $n$-list having a 1 at position $x_1$, \ldots, $x_k$, and a 0 elsewhere.

Example: 2-combinations of $\{1,2,3,4\}$: $\{1,2\}$, $\{1,3\}$, $\{1,4\}$, $\{2,3\}$, $\{2,4\}$, $\{3,4\}$. Corresponding 4-lists of $\{0,1\}$: $1100$, $1010$, $1001$, $0110$, $0101$, $0011$.

The $n$-lists of $\{0,1\}$ containing $k$ times 1, and therefore equivalently the $k$-combinations of $n$-elements of $S$, can be generated in lexicographic order with an algorithm that builds the $k$-list of position $t+1$ using only the $k$-list of position $t$, and which stops without counting the combinations generated. For example, the MATLAB function NCHOOSEK(S,k), where $S$ is the row vector of length $n$ of the elements of $S$, creates in lexicographic order a $k$ columns matrix whose rows consist of all possible combinations of the $n$ elements of $S$ taken $k$ at a time. The number of such combinations, given by the binomial coefficient $n!/((n-k)! k!)$, can be also computed with the function NCHOOSEK by replacing the first argument, the row vector $S$, with the scalar $n$.

Unfortunately the binomial coefficient increases rapidly with $n$, which makes the generation of all $k$-combinations computationally hard: with NCHOOSEK the task is impractical even for values just above 15. However, a lexicographic algorithm implements a one-to-one correspondence between the $k$-combinations and the generation order, i.e. the set of numbers $s = 1,\ldots,(n!/((n-k)!k!))$. This fact is used in our function to determine the $n$-list corresponding to the $k$-combination $\{x_1, \ldots, x_k\}$ which would be generated by the lexicographic algorithm at a given desired position $N$. This is useful in a number of applications which require one or several, but not all, $k$-combinations at given generation order(s).