Our main result here is that the specialization at
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si1.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=bfb7cfcdb14e8aaf95e847d1ef0cf7a1" title="Click to view the MathML source">t=1/qclass="mathContainer hidden">class="mathCode"> of the
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si2.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=d120c0b33398e3f66366b0b7fc128357" title="Click to view the MathML source">Qkm,knclass="mathContainer hidden">class="mathCode"> operators studied in Bergeron et al.
[2] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si1.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=bfb7cfcdb14e8aaf95e847d1ef0cf7a1" title="Click to view the MathML source">t=1/qclass="mathContainer hidden">class="mathCode"> to the Rational Compositional Shuffle conjecture of Bergeron et al.
[3]. In particular we show that if
m,
n and
k are positive integers and
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si3.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=ae30d03eead71033e8dd933c78819d88" title="Click to view the MathML source">(m,n)class="mathContainer hidden">class="mathCode"> is a coprime pair then
class="formula" id="fm0010">
where as customarily, for any integer
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si177.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=8bb5b340ddf7713ce721dd4c8dc45a37" title="Click to view the MathML source">s≥0class="mathContainer hidden">class="mathCode"> and indeterminate
u we set
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si6.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=a73edd762f790ee2e9079d30254e214e" title="Click to view the MathML source">[s]u=1+u+⋯+us−1class="mathContainer hidden">class="mathCode">. We also show that the symmetric polynomial on the right hand side is always Schur positive. Moreover, using the Rational Compositional Shuffle conjecture, we derive a precise formula expressing this polynomial in terms of Parking Functions in the
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300541&_mathId=si69.gif&_user=111111111&_pii=S0097316516300541&_rdoc=1&_issn=00973165&md5=3ed99bf7ae48efaa5340eedea6950942" title="Click to view the MathML source">km×knclass="mathContainer hidden">class="mathCode"> lattice rectangle.