Counting permutations by the number of successions within cycles

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• 作者：Toufik Mansour^a ; ^{tmansour@univ.haifa.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Mark Shattuck^b ; ^{shattuck@math.utk.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Succession ; Permutation ; Stirling number of first kind ; Combinatorial proof
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 April 2016
• 年：2016
• 卷：339
• 期：4
• 页码：1368-1376
• 全文大小：397 K

文摘

In this note, we consider the problem of counting (cycle) successions  , i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004379&_mathId=si1.gif&_user=111111111&_pii=S0012365X15004379&_rdoc=1&_issn=0012365X&md5=ad3c621ab97c3fbe0dbbf37bb476543c" title="Click to view the MathML source">S_nclass="mathContainer hidden">class="mathCode">${\mathcal{S}}_n$ for the statistics recording the number of cycles and adjacencies of the form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004379&_mathId=si2.gif&_user=111111111&_pii=S0012365X15004379&_rdoc=1&_issn=0012365X&md5=505890af31927fdb336d36311c23f6d7" title="Click to view the MathML source">j,j+dclass="mathContainer hidden">class="mathCode">$j,j+d$ where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004379&_mathId=si3.gif&_user=111111111&_pii=S0012365X15004379&_rdoc=1&_issn=0012365X&md5=364c767fdd466b9a04fa83b01f1c1e6d" title="Click to view the MathML source">d>0class="mathContainer hidden">class="mathCode">$d>0$ which extends earlier results.