TY - JOUR

T1 - Weyl group q-Kreweras numbers and cyclic sieving

AU - Reiner, Victor

AU - Sommers, Eric

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A,B,C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A,B,C,D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

AB - Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A,B,C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A,B,C,D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

KW - Catalan number

KW - Cyclic sieving phenomenon

KW - Kreweras number

KW - Narayana number

KW - Nilpotent orbit

KW - Reflection

KW - Weyl group

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U2 - 10.1007/s00026-018-0408-y

DO - 10.1007/s00026-018-0408-y

M3 - Article

AN - SCOPUS:85063614872

VL - 22

SP - 819

EP - 874

JO - Annals of Combinatorics

JF - Annals of Combinatorics

SN - 0218-0006

IS - 4

ER -