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Article overview
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A three shuffle case of the compositional parking function conjecture | | Adriano M. Garsia
; Guoce Xin
; Mike Zabrocki
; | | Date: |
28 Aug 2012 | | Abstract: | We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q,
t-enumerates, by the statistics dinv and area, the parking functions whose
supporting Dyck path touches the main diagonal according to the composition p
of size a + b + c and have a reading word which is a shuffle of one decreasing
word and two increasing words of respective sizes a, b, c. Here Cp(1) is a
rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator
introduced in [1]. This is our latest progress in a continued effort to settle
the decade old shuffle conjecture of [14]. It includes as special cases all
previous results connected with this conjecture such as the q, t-Catalan [3]
and the Schroder and h, h results of Haglund in [12] as well as their
compositional refinements recently obtained in [9] and [10]. It also confirms
the possibility that the approach adopted in [9] and [10] has the potential to
yield a resolution of the shuffle parking function conjecture as well as its
compositional refinement more recently proposed by Haglund, Morse and Zabrocki
in [15]. | | Source: | arXiv, 1208.5796 | | Services: | Forum | Review | PDF | Favorites | |
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