6,106 research outputs found
Sweep maps: A continuous family of sorting algorithms
We define a family of maps on lattice paths, called sweep maps, that assign
levels to each step in the path and sort steps according to their level.
Surprisingly, although sweep maps act by sorting, they appear to be bijective
in general. The sweep maps give concise combinatorial formulas for the
q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers,
and many more general polynomials connected to the nabla operator and rational
Catalan combinatorics. We prove that many algorithms that have appeared in the
literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa,
Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the
present authors) are all special cases of the sweep maps or their inverses. The
sweep maps provide a very simple unifying framework for understanding all of
these algorithms. We explain how inversion of the sweep map (which is an open
problem in general) can be solved in known special cases by finding a "bounce
path" for the lattice paths under consideration. We also define a generalized
sweep map acting on words over arbitrary alphabets with arbitrary weights,
which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac
Evaluating the efficacy of a GP led pre diabetes intervention targeting lifestyle modification
Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials
We introduce explicit combinatorial interpretations for the coefficients in
some of the transition matrices relating to skew Hall-Littlewood polynomials
P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials
G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the
Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials
M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak
quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of
the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing
starred tableaux.Comment: 28 pages; added brief discussion of the Hall-Littlewood Q', typos
corrected, added references in response to referee suggestion
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