This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The a(8) = 6 set partitions:
{{1,2,3,4,5,6,7,8}}
{{1,2,7,8},{3,4,5,6}}
{{1,3,6,8},{2,4,5,7}}
{{1,4,5,8},{2,3,6,7}}
{{1,4,6,7},{2,3,5,8}}
{{1,8},{2,7},{3,6},{4,5}}
a(3) = 1: 12|3 a(4) = 3: 12|3|4, 13|2|4, 14|23. a(5) = 9: 12|3|45, 12|3|4|5, 13|25|4, 13|2|4|5, 14|23|5, 1|23|4|5, 14|2|3|5, 15|24|3, 1|25|34.
The a(3) = 77 set-systems:
123 1-23 1-2-3 1-2-3-13 1-2-3-13-23 1-2-3-13-23-123
2-13 1-2-13 1-2-3-23 1-2-12-13-23 1-2-12-13-23-123
1-123 1-2-23 1-2-12-13 1-2-3-13-123
12-13 1-3-23 1-2-12-23 1-2-3-23-123
12-23 2-3-13 1-2-13-23 1-2-12-13-123
13-23 1-12-13 1-2-3-123 1-2-12-23-123
2-123 1-12-23 1-3-13-23 1-2-13-23-123
3-123 1-13-23 2-3-13-23 1-3-13-23-123
12-123 1-2-123 1-12-13-23 2-3-13-23-123
13-123 1-3-123 1-2-12-123 1-12-13-23-123
23-123 2-12-13 1-2-13-123 2-12-13-23-123
2-12-23 1-2-23-123
2-13-23 1-3-13-123
2-3-123 1-3-23-123
3-13-23 2-12-13-23
1-12-123 2-3-13-123
1-13-123 2-3-23-123
12-13-23 1-12-13-123
1-23-123 1-12-23-123
2-12-123 1-13-23-123
2-13-123 2-12-13-123
2-23-123 2-12-23-123
3-13-123 2-13-23-123
3-23-123 3-13-23-123
12-13-123 12-13-23-123
12-23-123
13-23-123
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
qes[n_]:=Select[stableSets[Subsets[Range[n],{1,n}],Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[qes[n]],{n,0,4}]
\\ by inclusion/exclusion on covered vertices.
C(v)={my(u=Vecrev(-1 + prod(k=1, #v, 1 + x^v[k]))); prod(i=1, #u, 1 + u[i])}
a(n)={my(s=0); forsubset(n, v, s += (-1)^(n-#v)*C(v)); s} \\ Andrew Howroyd, Oct 02 2019
The graph with edge-set {{1,2},{1,3},{1,4},{2,3}}, which looks like a triangle with a tail, has edges {1,4} and {2,3} with equal sum, so is not counted under a(4).
a:= proc(n) option remember; `if`(n=0, 1,
a(n-1)*ceil(n/2)*ceil(n/2+1/4))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Oct 03 2019
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
qes[n_]:=stableSets[Subsets[Range[n],{2}],Total[#1]==Total[#2]&];
Table[Length[qes[n]],{n,0,5}]
a(n) = {prod(k=1, 2*n+1, ceil(k/4))} \\ Andrew Howroyd, Oct 02 2019
The a(n) set partitions for n = 3, 7, 11, 15, 23:
{12} {123} {124} {1234} {1235}
{1}{2} {1}{23} {1}{24} {1}{234} {1}{235}
{13}{2} {12}{4} {12}{34} {12}{35}
{1}{2}{3} {14}{2} {123}{4} {123}{5}
{1}{2}{4} {124}{3} {125}{3}
{13}{24} {13}{25}
{134}{2} {135}{2}
{1}{2}{34} {15}{23}
{1}{23}{4} {1}{2}{35}
{1}{24}{3} {1}{25}{3}
{14}{2}{3} {13}{2}{5}
{1}{2}{3}{4} {15}{2}{3}
{1}{2}{3}{5}
The a(1) = 1 through a(5) = 12 set partitions:
{{1}} {{12}} {{123}} {{1234}} {{12345}}
{{1}{2}} {{13}{2}} {{12}{34}} {{1245}{3}}
{{1}{2}{3}} {{13}{24}} {{135}{24}}
{{14}{23}} {{15}{234}}
{{1}{23}{4}} {{1}{234}{5}}
{{14}{2}{3}} {{12}{3}{45}}
{{1}{2}{3}{4}} {{135}{2}{4}}
{{14}{25}{3}}
{{15}{24}{3}}
{{1}{24}{3}{5}}
{{15}{2}{3}{4}}
{{1}{2}{3}{4}{5}}
The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).
The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(5).
The a(8) = 10 rooted twice-partitions: (6), (51), (42), (321), (5)(), (41)(), (32)(), (4)(1), (3)(2), (3)(1)().
twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],And[UnsameQ@@Total/@#,UnsameQ@@Join@@#]&]//Length,{n,20}]
The a(1) = 1 through a(4) = 9 set partitions:
{{1}} {{12}} {{123}} {{1234}}
{{1}{2}} {{13}{2}} {{12}{34}}
{{1}{2}{3}} {{124}{3}}
{{13}{24}}
{{134}{2}}
{{14}{23}}
{{1}{23}{4}}
{{14}{2}{3}}
{{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).
Comments