Enumerative Combinatorics

Abstract

Combinatorics. Sets, Multisets, Sieve Methods, Ferrers Board, posets, New posets from old, lattice theory, Incidence Algebra of a Locally finite poset, Rank selection, Rational gnerating functions. page 96-201: Exclusion-Inclusion principle has somewhat to do with posets and this with Moebius function. Theory of binomial posets. Hasse diagram (a brief description how to draw a Hasse diagram). Isomorphisms. Graded posets: If every maximal chain in P has the same length n then we say that P is graded of rank n. In this case there is a unique rank function rho: P--> {0,1,...,n} such that rho(x)=0 if x is a minimal element of P and rho(y)=rho(x) + 1 if y covers x in P. If rho(x)=i then we say that x has rank i. If P is graded of rank n and has p(i) elements of rank i then the polynomial F(P,q) = sum p(i)*qi i=0 to n is called the rank generating function of P. If graded partial order sets are combined by cartesian products, the resulting pasrtial order is also graded and the Polynom can be obteined just by polynomial multiplication of the single polynoms. The coefficients of the powers , say power k say how many elements are having the rank k. Note that the ranks are coiunted from the bottom with 0. Order ideals and order filters. J(P) the set of all order ideals, ordered by inclusion. ** Erw ** 2.11.06: J(P) is a distributive lattice. Joint irreducibles with induced order are isomorphic to a poset. series-parallel posets: disjoint union and ordinal sum. Page 107: If P is an n-element poset, then J(P) is graded of rank n. The vertices of J(P): The rank is equal to the number of elements of the ideal. Linear extensions of P. "The number of extensions of P to a total order is denoted as e(P) and is probably the single most useful number for measuring the "complexity " of P." e(P) is the number of maximal chains of J(P). Certain lattice path problems are equivalent to determining e(P). Disjoint sums of partial orders P(i): P=sum P(i) then e(P) = (n1+..nk)!/[(n1!)(n2)!...(nk)!] * e(P1)*e(P2)*...*e(Pk) The ni are the cardinalities of the i th poset. k posets are summed. Standort: Zuhause ##20Z##