Stirling Number of the First Kind Calculator

About this calculator

The unsigned Stirling number of the first kind, c(n,k), counts the number of ways to arrange n distinct elements into exactly k cycles — closed loops of permutation, as opposed to fixed sequences. They satisfy the recurrence c(n,k) = c(n−1,k−1) + (n−1)·c(n−1,k), building up from the base cases c(0,0)=1 and c(n,0)=0 for n greater than 0.

This calculator computes c(n,k) directly for the n and k you enter. Stirling numbers of the first kind show up in combinatorics, permutation group theory, and the study of algorithms that analyze cycle structure, and they're the coefficients that relate falling factorials to ordinary powers — a useful reference for students and researchers working through combinatorial identities.