EPISODE · Jul 15, 2025 · 5 MIN
OEIS A000276: Two-Cycle Permutations With No Fixed Points
from Intellectually Curious · host Mike Breault
In this episode we explore A000276, the associated Stirling numbers of the first kind that count permutations of n with no fixed points and exactly two cycles. We unpack the defining count and the key formula a_n = n! × sum_{k=2}^{n-2} (1/k), showing how these integers refine permutation cycle structure. We’ll see how traditional tables look like stair-steps, and how a linear transformation reshapes them into a Pascal-like arithmetical triangle, revealing hidden order. We’ll also note that, unlike many Stirling numbers of the second kind, these do not form a Newton–Euler sequence, highlighting their distinctive divisibility and congruence behavior. Finally, we discuss applications in combinatorics and graph theory where counting cycle configurations with no fixed points matters, illustrating why these numbers matter beyond pure theory.Note: This podcast was AI-generated, and sometimes AI can make mistakes. Please double-check any critical information.Sponsored by Embersilk LLC
What this episode covers
In this episode we explore A000276, the associated Stirling numbers of the first kind that count permutations of n with no fixed points and exactly two cycles. We unpack the defining count and the key formula a_n = n! × sum_{k=2}^{n-2} (1/k), showing how these integers refine permutation cycle structure. We’ll see how traditional tables look like stair-steps, and how a linear transformation reshapes them into a Pascal-like arithmetical triangle, revealing hidden order. We’ll also note that, u...
NOW PLAYING
OEIS A000276: Two-Cycle Permutations With No Fixed Points
No transcript for this episode yet
Similar Episodes
No similar episodes found.