Another Useful Representation of Stirling Number for Biology

Adrian Kllogjeri¹ & Pellumb Kllogjeri^2*

¹Department of Applied Econometrics, Risk Modeller at ZPG, London, United Kingdom
²Doctor of Philosophy in Mathematics and Computer Sciences, Statistics and Graph Theory, University Aleksander Xhuvani, Department of Mathematics, Elbasan, Albania

Dr. Pellumb Kllogjeri, Doctor of Philosophy in Mathematics and Computer Sciences, Statistics and Graph Theory, University Aleksander Xhuvani, Department of Mathematics, Elbasan, Albania.

Keywords: Partition of a Set; Partition of Number n in Accordance with Its Composition with Summands; Classes; Mathematical Biology; 4-Letter Alphabet; Protein Sequence

Abstract

In the paper “Partition of a Set with N Elements into K Blocks with Number of Elements in Accordance with the Composition of Number N As a Sum of Any K Natural Summands (Another Representation of Stirling Number)”- published in the International Journal of Advanced Computing [1], we have presented a new formula for computing the number of partitions of the set [n] into k blocks with respect to the representation of the number n as a sum of k natural summands (different or equal). The Stirling numbers of the second kind describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets.
The study is finalized with an extraordinary identity between the second type Stirling number and the number of partitions of number n in accordance with the composition of number n with summands. The new representation of Stirling numbers is very useful in biology for computing the number of partitions of number n which is the same result achieved by Stirling. Our paper is a specific contribution for specific needs, particularly related to the computational problems in biology. Here will find answer all of those, researching in different fields of life and science, who are concerned about the number of partitions of number n into k parts with the same number of elements, with different number of elements or, several parts having the same number of elements and the others different numbers of elements.

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