A284276
Number of event structures with n labeled elements.
Original entry on oeis.org
1, 4, 41, 916, 41099, 3528258, 561658287
Offset: 1
An event structure is given by a poset and a conflict relation (denoted #) on it. The conflict relation is irreflexive and symmetric, and must propagate over the order: a<=b and a#c imply b#c.
For n=2, (i.e., two elements a and b), there are three possible posets: a<=b, b<=a, and neither of the two. For the first two cases, only the empty conflict is possible. For the third case, you can have either the empty conflict relation, or a#b. Hence the total number of event structures is 4.
- Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
- Marco B. Caminati, Isabelle/HOL code
- Marco B. Caminati and Juliana K. F. Bowles, Representation Theorems Obtained by Mining across Web Sources for Hints, Lancaster Univ. (UK, 2022).
- M. Nielsen, G. Plotkin, and G. Winskel, Petri nets, event structures and domains, part I, Theoretical Computer Science 13, no. 1 (1981): 85-108.
- G. Winskel and M. Nielsen, Models for concurrency, DAIMI Report Series 21, no. 429 (1992) (revised version).
Cf.
A001035 (generating all the event structures entails generating all the posets),
A000798 (to generate all the posets we preemptively generated all the quasi-orders).
A284762
Total number of subsets of X that are open and closed and connected summed over all distinct topological spaces X that can be placed on an n-set.
Original entry on oeis.org
1, 2, 9, 69, 852, 16363, 479435, 21150888, 1388124543, 133822887673, 18707633394606, 3745998552621317, 1062675319801676431, 423005074717335908762, 234301896939296139079453, 179277553685814268284430793, 188286118651948743843774496644, 269901723843412313246289232355847, 525443899393186663528068248469425039
Offset: 0
a(2) = 9. Let X = {a,b}. There are four distinct topologies (A000798) that can be placed on X: {{},X} {{},{a},X} {{}, {b},X} {{},{a},{b},X}. These topologies have 2 + 2 + 2 + 3 sets respectively that are open and closed and connected.
A376064
Number of quasi-orders on an n-set that are not partial orders.
Original entry on oeis.org
0, 0, 1, 10, 136, 2711, 79504, 3405382, 211055975, 18749246912, 2365988624260, 420564361630293, 104490620009920522, 36030665275081893690, 17132727719926060775277, 11169098098145556139435182, 9930583626219881751366237516, 11985408843042557809380587456695, 19553143146433198202168306753032180
Offset: 0
-
a[n_]:=Part[ResourceFunction["OEISSequence"]["A000798"],n+1]-Part[ResourceFunction["OEISSequence"]["A001035"],n+1]; Array[a,18,0] (* Stefano Spezia, Sep 08 2024 *)
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