Cyclic sieving on permutations: An analysis of maps and statistics in the FindStat database

We perform a systematic study of permutation statistics and bijective maps on permutations using SageMath to search the FindStat combinatorial statistics database to identify apparent instances of the cyclic sieving phenomenon (CSP). Cyclic sieving occurs on a set of objects, a statistic, and a map of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when the evaluation of the statistic generating function at the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power of the primitive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th root of unity equals the number of fixed points under the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power of the map. Of the apparent instances found in our experiment, we prove 34 new instances of the CSP and conjecture three more. Our results are organized largely by orbit structure, proving instances of the CSP for involutions with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n minus 1"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixed points and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript left floor StartFraction n Over 2 EndFraction right floor"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false"> ⌊ </mml:mo> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence="false" stretchy="false"> ⌋ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{\lfloor \frac {n}{2}\rfloor }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixed points, as well as maps whose orbits all have the same size. The FindStat maps which exhibit the CSP include a map constructed by Corteel (using a bijection of Foata and Zeilberger) to swap the number of nestings and crossings, the invert Laguerre heap map, a map of Alexandersson and Kebede designed to preserve right-to-left minima, conjugation by the long cycle, as well as reverse, complement, rotation, Lehmer code rotation, and toric promotion. Our results combined with prior work of the last 5 authors in (Math. Comp. <bold>93</bold> , 921–976) show that, contrary to common expectations, actions that exhibit homomesy are not necessarily the best candidates for the CSP, and vice versa.