Catalan Path Statistics having the Narayana Distribution
This preprint investigates determining the statistics on the Catalan lattice
paths satisfying the Narayana distribution. It extends the list of known
statistics considerably and shows how the statistics relate to one another
by moderately simple bijections.
Guessing, Ballot Numbers, and Refining Pascal's Triangle
(1995) Given a shuffled deck of playing cards, consider drawing the top card, guessing
its color, and discarding it face up. While always knowing the numbers
of cards of each color remaining, repeat this until the deck is exhausted.
If one does not guess deviously, what is the probability distribution for
the number of wrong guesses? Using lattice path notions, we find one such distribution,
which is related to the refinement of the binomial coefficients
as lists of ballot numbers. This solution leads
to a refinement of Pascal's triangle.
Comments on this draft are very welcome. Again, `thanks' to Curtis Mack.