Signed Stirling numbers

Definition

Signed Stirling numbers, denoted as $s(n, k)$, are combinatorial numbers that count the number of permutations of $n$ elements with exactly $k$ cycles, considering the sign of the permutation. These numbers are closely related to the Stirling numbers of the first kind, which count the same cycles without considering their signs. The signed version provides a way to account for the parity of the permutations, distinguishing between even and odd permutations.

5 Must Know Facts For Your Next Test

1. The signed Stirling number can be calculated using the recurrence relation: $s(n, k) = s(n-1, k-1) - (n-1) s(n-1, k)$.
2. The signed Stirling numbers can be used to derive properties related to alternating permutations and their connections to other combinatorial structures.
3. For a given $n$, the sum of all signed Stirling numbers $s(n, k)$ over all $k$ equals zero: $\sum_{k=0}^{n} s(n, k) = 0$.
4. Signed Stirling numbers can also be expressed in terms of generating functions, providing insights into their combinatorial significance.
5. These numbers have applications in various areas including algebraic combinatorics and representation theory.

Review Questions

• How do signed Stirling numbers differ from traditional Stirling numbers of the first kind?
  • Signed Stirling numbers differ from traditional Stirling numbers by accounting for the sign or parity of permutations. While Stirling numbers of the first kind count permutations with a certain number of cycles without regard to their signs, signed Stirling numbers specifically count these permutations while considering whether they are even or odd. This distinction allows signed Stirling numbers to provide additional insights into the structure of permutations.
• Discuss how the recurrence relation for signed Stirling numbers reflects their combinatorial interpretation.
  • The recurrence relation for signed Stirling numbers, $s(n, k) = s(n-1, k-1) - (n-1) s(n-1, k)$, reflects their combinatorial interpretation by showing how to build permutations incrementally. The first term $s(n-1, k-1)$ corresponds to adding a new element that forms a new cycle, while the second term accounts for adding this new element into existing cycles. This relationship illustrates how the signed nature of these counts interacts with the structure of permutations and cycle formation.
• Evaluate how signed Stirling numbers connect to other areas such as algebraic combinatorics and representation theory.
  • Signed Stirling numbers play a significant role in algebraic combinatorics and representation theory by providing tools to study symmetric functions and character theory. Their relationships with permutations and cycle structures facilitate the exploration of generating functions that encapsulate combinatorial identities. Additionally, they help in understanding how different representations interact through alternating groups, revealing deep connections between combinatorial objects and algebraic structures within these fields.

Related terms

Stirling numbers of the first kind: These are a type of combinatorial number that counts the number of permutations of $n$ elements with $k$ cycles, without taking into account the sign of the permutations.

Permutation: A rearrangement of a set of elements into a sequence or order, where the order matters and different arrangements are considered distinct.

Cycle decomposition: A way to express a permutation as a product of disjoint cycles, where each cycle represents a closed path among elements in the permutation.