scholarly journals Stirling Permutations, Cycle Structure of Permutations and Perfect Matchings

10.37236/5318 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Shi-Mei Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Perfect Matchings ◽  
Maximal Elements ◽  
Cycle Structure ◽  
Constructive Proofs

In this paper we provide constructive proofs that the following three statistics are equidistributed: the number of ascent plateaus of Stirling permutations of order $n$, a weighted variant of the number of excedances in permutations of length $n$ and the number of blocks with even maximal elements in perfect matchings of the set $\{1,2,3,\ldots,2n\}$.

scholarly journals Eulerian Polynomials, Stirling Permutations of the Second Kind and Perfect Matchings

10.37236/7288 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Shi-Mei Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Fixed Points ◽  
Perfect Matchings ◽  
Eulerian Polynomials ◽  
Constructive Proofs

In this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary derangement polynomials in terms of the Stirling derangement polynomials. Finally, we present constructive proofs of a kind of combinatorial expansions of the Eulerian polynomials of types $A$ and $B$.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

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