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Enumeration of Chains in Semi-Primary Lattices
F. Regonati1, S.D. Sarti2
1University of Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
regonati@dm.unibo.it
2ITCG Paolini, via Gucciardini 2, 40026 Imola, Italy
Annals of Combinatorics 4 (1) p.109-124 March, 2000
AMS Subject Classification:05A15, 05A30, 06A08, 06C05
Abstract:
We show that some combinatorial properties of vector spaces (see [13]) and finite abelian p-groups (see [5, 6]) may be carried over to the purely synthetic setting of q-primary lattices (see [1,12]).
Let L be a q-primary lattice and c: 0 = c0 < c1 < #< cn = 1 a maximal chain such that the join of all the upper covers of any term of c is still a term of c, and for each x L, consider the subset of [n] (see [16]) S(x) = {i∈[n]; ci-1 x < cix}.
Our main results are:
  • the sets S(x); x L, form, with respect to set union and intersection, a 1-primary lattice L', i.e., a direct product of chains, and the mapping S is order- and type-preserving;
  • if X is any chain in L' and w = w1w2wn is the labeling of [n] induced by X, then the number of chains in L which are sent by S to X is qinv(w); where the exponent counts the ※minimal inversions§of w.
Our proof is based on the properties of radicals and socles in q-primary lattices [7, 21].
Keywords: semi-primary lattice, q每primary lattice, flag每victors, minimal inversions, Hall polynomials

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