A pattern $P$ of length $j$ has the minimal overlapping property if two consecutive occurrences of the pattern can overlap in at most one place, namely, at the end of the first consecutive occurrence of the pattern and at the start of the second consecutive occurrence of the pattern. For patterns $P$ which have the minimal overlapping property, we derive a general formula for the generating function for the number of consecutive occurrences of $P$ in words, permutations and $k$-colored permutations in terms of the number of maximum packings of $P$ which are patterns of minimal length which has $n$ consecutive occurrences of the pattern $P$. Our results have as special cases several results which have appeared in the literature. Another consequence of our results is to prove a conjecture of Elizalde that two permutations $\alpha$ and $\beta$ of size $j$ which have the minimal overlapping property are strongly $c$-Wilf equivalent if $\alpha$ and $\beta$ have the same first and last elements.
Abstract. In a recent paper [8] J. Haglund showed that the expression ∆ h j E n,k , en with ∆ h j the Macdonald eigen-operator ∆ h jH µ = hj[Bµ]Hµ enumerates by t area q dinv the parking functions whose diagonal word is in the shuffle 12 · · · j∪∪j + 1 · · · j + n with k of the cars j + 1, . . . , j + n in the main diagonal including car j + n in the cell (1, 1). In view of some recent conjectures of Haglund-Morse-Zabrocki [12] it is natural to conjecture that replacing E n,k by the modified Hall-Littlewood funtions Cp 1 Cp 2 · · · Cp k 1 would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j + 1, . . . , j + n hits the diagonal according to the composition p = (p1, p2, . . . , p k ). We prove here this conjecture by deriving a recursion for the polynomial ∆ h j Cp 1 Cp 2 · · · Cp k 1 , en then using this recursion to construct a new dinv statistic we will denote ndinv and show that this polynomial enumerates the latter parking functions by t area q ndinv
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