A Lah number counts how many ways you can divide n different things into k ordered groups. The order within a group plays a role here, but the order of the groups does not. Lah numbers were first described in 1955 by the Slovenian mathematician and actuary Ivo Lah.
The following property is mathematically more abstract: The Lah numbers describe the correlation coefficients between increasing and decreasing factorials. An increasing factorial is calculated from bottom to top, i.e., 1*2*3*...*n, and a decreasing factorial is calculated from top to bottom, i.e., n*(n-1)*...*1. If you want to express an increasing factorial in terms of decreasing factorials, or vice versa, you need the corresponding Lah numbers.
| Ln,k = | ( | n-1 | ) | n! |
| k-1 | k! |
Enter for n and k positive integers. n must be larger as k. Then press Calculate to determine the Lah number.
An example: A group of twelve people is divided into three teams. The teams do not have to be the same size; any combination is allowed, but empty teams are prohibited; a team must have at least one member. The number of elements per group, i.e., people per team, is not needed for this calculation. So, n=12 and k=3. Each person in each team must give a presentation within the team; the different teams are independent of each other and indistinguishable. The teams therefore have no order, but the members within each team do. If the teams were distinguishable, then you would have to multiply by 3!, which is 6.
The Lah number L12,3 indicates how many possible orders there are for the presentations. This is 4,390,848,000, or just under 4.4 billion. This number is so high because, of course, there are many ways to divide twelve people into three teams. This value is given by the Stirling number, the calculation of which is more complicated than the Lah number. The number is then increased again by the different possible orders within the individual groups.
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