Lattice path moments and the cut shuffle

E. Pergola, R. Pinzani, S. Rinaldi, R.A. Sulanke

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


In the integer coordinate plane we consider those lattice paths whose step types consists of (1,1), (1,-1), and perhaps some horizontal steps. For those paths running from (0,0) to (n+2,0) and remaining strictly above the x-axis elsewhere, we will define a zeroth moment(the cardinality), a first moment (the total area), and a second moment, in terms of the lattice points traced by its paths. We will define a bijection relating these moments to the cardinalities of sets of selected lattice points on the unrestricted paths running from (0,0) to (n,0). This bijection acts by cutting the elevated paths and then shuffling the resulting subpaths by a fixed permutation. For cardinality, this ``cut shuffling'' is a variant of the well-known cycle lemma.

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