570 research outputs found
A Generalization of the Ramanujan Polynomials and Plane Trees
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has
recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1,
Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's
conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and
answer his question about the combinatorial interpretation of Q_n. Actually we
give combinatorial interpretations of these polynomials in terms of plane
trees, half-mobile trees, and forests of plane trees. Our approach also leads
to a general formula that unifies several known results for enumerating trees
and plane trees.Comment: 20 pages, 2 tables, 8 figures, see also
http://math.univ-lyon1.fr/~gu
Pattern Avoidance in Task-Precedence Posets
We have extended classical pattern avoidance to a new structure: multiple
task-precedence posets whose Hasse diagrams have three levels, which we will
call diamonds. The vertices of each diamond are assigned labels which are
compatible with the poset. A corresponding permutation is formed by reading
these labels by increasing levels, and then from left to right. We used Sage to
form enumerative conjectures for the associated permutations avoiding
collections of patterns of length three, which we then proved. We have
discovered a bijection between diamonds avoiding 132 and certain generalized
Dyck paths. We have also found the generating function for descents, and
therefore the number of avoiders, in these permutations for the majority of
collections of patterns of length three. An interesting application of this
work (and the motivating example) can be found when task-precedence posets
represent warehouse package fulfillment by robots, in which case avoidance of
both 231 and 321 ensures we never stack two heavier packages on top of a
lighter package.Comment: 17 page
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