Construct a combinatorial shifted analogue of parking functions

Construct a combinatorial model of shifted parking functions of length n whose Frobenius characteristic is the shifted parking function symmetric function SH_n = PF_n(x/x), where PF_n is the Frobenius characteristic of the natural S_n-action on ordinary parking functions. Ideally, develop this model so that it aligns with the structural properties suggested in Section 4, such as a block decomposition indexed by partitions of n into odd parts with block sizes matching the coefficients in the Schur P-basis polynomial expansion of SH_n given by Theorem 3.4.

Background

The paper defines SH_n as the shiftification of the parking function symmetric function PF_n, namely SH_n(x) = PF_n(x/x). It develops explicit expansions of SH_n in terms of power sums, Schur P-functions, and as a polynomial in the algebraically independent P_{2i+1}’s. These results parallel classical expansions for PF_n and establish enumerative identities such as dim SH_n = 2^n (n + 1)^{n−1}.

However, the authors emphasize that a combinatorial object corresponding to SH_n—analogous to ordinary parking functions for PF_n—is missing. In Section 4, they outline desirable features for such a “shifted parking function” model: there should be a natural partition of the objects into blocks indexed by partitions of n into odd parts, and the size of each block should equal the coefficient of the product P_{λ1}P_{λ2}... in the expansion of SH_n as a polynomial in the odd-indexed Schur P-functions (Theorem 3.4). They also discuss a naive model (colorings of ordinary parking functions) that matches some counts but lacks the desired structural correspondence, highlighting the need for a more faithful combinatorial construction.