TamariAnti

Contains the codes and formal context files for counting of (maximal) antichains of Tamari lattices

This is a supporing file for OEIS on computation of (maximal) antichains in the Tamari lattice of bracketing structures.

Outline

0. The basic code with the class for managing generation of formal concepts via NextClosure algorithm

1. Loading formal contexts for antichains and maximal antichains of Tamari lattice

2. Counting of maximal antichains for n in {0, 1, 2, 3, 4, 5, 6}

3. Counting of antichains for n in {0, 1, 2, 3, 4, 5}

Comment to Sections 2 and 3.

• We start with n=0 (i.e., with the set partition of zero elements), but in OEIS the respective offset is 1, i.e. there the sequences start with n=1. Note also that the n-th order of the Tamari lattice means that it is built over brackeing structures on n+1 elements.

• See folder AC_Tamari6 for counting antichains (ACs) in case n=6.

References

• Bernhard Ganter, Rudolf Wille: Formal Concept Analysis - Mathematical Foundations. Springer 1999, ISBN 978-3-540-62771-5, pp. I-X, 1-284.

• Tamari, Dov (1962), "The algebra of bracketings and their enumeration", Nieuw Archief voor Wiskunde, Series 3, 10: 131–146, MR 0146227.

• Huang, Samuel; Tamari, Dov (1972), "Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law", Journal of Combinatorial Theory, Series A, 13: 7–13, doi:10.1016/0097-3165(72)90003-9, MR 0306064.

• Knuth, Donald E. (2005), "Section 7.2.1.6: Generating All Trees", The Art of Computer Programming, vol. IV.

NB. Any usage of the codes requires the acknowledgment of this repository.