Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=d2003ff42ff32fb97c36010b77da6302" title="Click to view the MathML source">Snclass="mathContainer hidden">class="mathCode"> denote the set of permutations of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si2.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=64faeaccf5d84375f90741c8eaf76c56" title="Click to view the MathML source">[n]={1,2,…,n}class="mathContainer hidden">class="mathCode">. For a positive integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si3.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=1a5bf36b71a1b1ba2bcbc8b5e1121840" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">, define class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si4.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=1a347d2d52d9c56e8c037d165d26acd7" title="Click to view the MathML source">Sn,kclass="mathContainer hidden">class="mathCode"> to be the set of all permutations of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si5.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=12b2f8e53d7685521151fb6ed3404e2d" title="Click to view the MathML source">[n]class="mathContainer hidden">class="mathCode"> with exactly class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si3.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=1a5bf36b71a1b1ba2bcbc8b5e1121840" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode"> disjoint cycles, i.e.,
class="formula" id="fd000005">
class="mathml">class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=aa04edcb59cc17f00b3cf93bcb8b75d7" title="Click to view the MathML source">Sn,k={π∈Sn:π=c1c2⋯ck},class="mathContainer hidden">class="mathCode">class="temp" src="/sd/blank.gif">
where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si8.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=456682b4992ad0a18617267e1f2008b6" title="Click to view the MathML source">c1,c2,…,ckclass="mathContainer hidden">class="mathCode"> are disjoint cycles. The size of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si4.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=1a347d2d52d9c56e8c037d165d26acd7" title="Click to view the MathML source">Sn,kclass="mathContainer hidden">class="mathCode"> is given by class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si10.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=da73f17cbd6c2b1a8cf7477bf1470424">class="imgLazyJSB inlineImage" height="20" width="137" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15003623-si10.gif">class="mathContainer hidden">class="mathCode">, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si11.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ef647eb0b09b411e869b90010185fa41" title="Click to view the MathML source">s(n,k)class="mathContainer hidden">class="mathCode"> is the Stirling number of the first kind. A family class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si12.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=8ac7cd2b2c161b88ff9ed5580d0b6b82" title="Click to view the MathML source">A⊆Sn,kclass="mathContainer hidden">class="mathCode"> is said to be class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode">-cycle-intersecting if any two elements of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si14.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ecde62cae59304416b0c143daf55fdd7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode"> have at least class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode"> common cycles. A family class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si12.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=8ac7cd2b2c161b88ff9ed5580d0b6b82" title="Click to view the MathML source">A⊆Sn,kclass="mathContainer hidden">class="mathCode"> is said to be trivially class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode">-cycle-intersecting if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si14.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ecde62cae59304416b0c143daf55fdd7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode"> is the stabiliser of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode"> fixed points, i.e., class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si14.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ecde62cae59304416b0c143daf55fdd7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode"> consists of all permutations in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si4.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=1a347d2d52d9c56e8c037d165d26acd7" title="Click to view the MathML source">Sn,kclass="mathContainer hidden">class="mathCode"> with some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode"> fixed cycles of length one. For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si23.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=68f849392fb52554ff8b0b24be204ab2" title="Click to view the MathML source">1≤j≤tclass="mathContainer hidden">class="mathCode">, let
For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si25.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=05f616d8844e5a7d49fd5b680b84e89b" title="Click to view the MathML source">t+1≤s≤kclass="mathContainer hidden">class="mathCode">, let
In this paper, we show that, given any positive integers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si27.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=3ba9a04d498d6f48f4603697f001261d" title="Click to view the MathML source">k,tclass="mathContainer hidden">class="mathCode"> with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si28.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=926186f1093301bed3490525798ba5cb" title="Click to view the MathML source">k≥2t+3class="mathContainer hidden">class="mathCode">, there exists an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si29.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=3177290ee5ffd6debfd291f638d3f749" title="Click to view the MathML source">n0=n0(k,t)class="mathContainer hidden">class="mathCode">, such that for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si30.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=c9138f7a405718dc62db7f3470e76730" title="Click to view the MathML source">n≥n0class="mathContainer hidden">class="mathCode">, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si12.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=8ac7cd2b2c161b88ff9ed5580d0b6b82" title="Click to view the MathML source">A⊆Sn,kclass="mathContainer hidden">class="mathCode"> is non-trivially class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si13.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ebbaf7c620583342e6391ef65bfd1b03" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode">-cycle-intersecting, then
class="formula" id="fd000020">
class="mathml">class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si33.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=9109b109bf85d8446bafeb60bc1e5fc5" title="Click to view the MathML source">|A|≤|B|,class="mathContainer hidden">class="mathCode">class="temp" src="/sd/blank.gif">
where class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si34.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=aad6297e9e7f8f9d8795f51dd7b45b7f">class="imgLazyJSB inlineImage" height="24" width="177" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15003623-si34.gif">class="mathContainer hidden">class="mathCode">. Furthermore, equality holds if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si14.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=ecde62cae59304416b0c143daf55fdd7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode"> is a conjugate of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si36.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=390c380029d594224e1cba3595b24648" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">, i.e., class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si37.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=f6985e1f3cc0f20df963b485c034940d" title="Click to view the MathML source">A=β−1Bβclass="mathContainer hidden">class="mathCode"> for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003623&_mathId=si38.gif&_user=111111111&_pii=S0012365X15003623&_rdoc=1&_issn=0012365X&md5=c07d6813d921c52d64ff5001d461827e" title="Click to view the MathML source">β∈Snclass="mathContainer hidden">class="mathCode">.