A278993
Number of simple chord diagrams with n chords, up to rotation.
Original entry on oeis.org
0, 1, 1, 4, 21, 176, 1893, 25030, 382272, 6604535, 127222636, 2702798537, 62778105236, 1582725739329, 43046433007765, 1256332883208474, 39165907107963273, 1298945495674093932, 45666536827274985585, 1696460750775267473762
Offset: 1
A278994
Number of simple chord diagrams with n chords, modulo all symmetries.
Original entry on oeis.org
0, 1, 1, 4, 18, 116, 1060, 13019, 193425, 3313522, 63667788, 1351700744, 31390695708, 791372281393, 21523271532811, 628166776833181, 19582955637428422, 649472761243051940, 22833268501579122332, 848230375982060558217
Offset: 1
A306419
Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.
Original entry on oeis.org
1, 1, 1, 1, 4, 11, 32, 99, 326, 1123, 4064, 15291, 59924, 242945, 1019584, 4409233, 19648674, 89938705, 422744384, 2035739041, 10039057524, 50610247483, 260704414816, 1370387233859, 7346982653702, 40131663286851, 223238920709024, 1263531826402891, 7273434344119460
Offset: 0
The a(1) = 1 through a(5) = 11 set partitions:
{{1}} {{1}{2}} {{1}{2}{3}} {{13}{24}} {{1}{24}{35}}
{{1}{24}{3}} {{13}{24}{5}}
{{13}{2}{4}} {{13}{25}{4}}
{{1}{2}{3}{4}} {{14}{2}{35}}
{{14}{25}{3}}
{{1}{2}{35}{4}}
{{1}{24}{3}{5}}
{{1}{25}{3}{4}}
{{13}{2}{4}{5}}
{{14}{2}{3}{5}}
{{1}{2}{3}{4}{5}}
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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]]]];
Table[Length[stableSets[Complement[Subsets[Range[n],{2}],Sort/@Partition[Range[n],2,1,1]],Intersection[#1,#2]!={}&]],{n,0,10}]
(* Second program: *)
CompoundExpression[
b[n_] := I^(1 - n) 2^((n - 1)/2) HypergeometricU[(1 - n)/2, 3/2, -1/2],
Join[{1, 1, 1}, Table[Sum[(-1)^k b[n - 2 k] n (n - 1 - k)!/(k! (n - 2 k)!), {k, 0, n/2}], {n, 3, 20}]]
] (* Eric W. Weisstein, Sep 02 2025 *)
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\\ here b(n) is A000085(n)
b(n) = {sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
a(n) = {if(n < 3, n >= 0, sum(k=0, n\2, (-1)^k*b(n-2*k)*n*(n-1-k)!/(k!*(n-2*k)!)))} \\ Andrew Howroyd, Aug 30 2019
A334059
Triangle read by rows: T(n,k) is the number of perfect matchings on {1, 2, ..., 2n} with k disjoint strings of adjacent short pairs.
Original entry on oeis.org
1, 0, 1, 1, 2, 0, 5, 8, 2, 0, 36, 49, 19, 1, 0, 329, 414, 180, 22, 0, 0, 3655, 4398, 1986, 344, 12, 0, 0, 47844, 55897, 25722, 5292, 377, 3, 0, 0, 721315, 825056, 384366, 87296, 8746, 246, 0, 0, 0, 12310199, 13856570, 6513530, 1577350, 192250, 9436, 90, 0, 0, 0
Offset: 0
Triangle begins:
1;
0, 1;
1, 2, 0;
5, 8, 2, 0;
36, 49, 19, 1 0;
...
For n=2 and k=1 the configurations are (1,4),(2,3) (i.e. a single short pair) and (1,2),(3,4) (i.e. two adjacent short pairs); hence T(2,1) = 2.
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CoefficientList[Normal[Series[Sum[y^j*(2*j)!/2^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(2*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
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T(n)={my(v=Vec(sum(j=0, n, (2*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(2*j+1) / (j! * 2^j * (1-(1-y)*x^2 + O(x*x^n))^(2*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 25 2020
A378862
Number of minimum edge covers in the n-cycle complement graph.
Original entry on oeis.org
0, 1, 5, 4, 70, 31, 972, 293, 14476, 3326, 237575, 44189, 4305960, 673471, 85836485, 11588884, 1871150248, 222304897
Offset: 3
A367000
Triangle read by rows: T(n,k) is the total number of bubbles of size k found in linear chord diagrams on 2n vertices.
Original entry on oeis.org
0, 0, 2, 0, 0, 1, 8, 4, 2, 2, 0, 5, 42, 30, 20, 15, 12, 10, 0, 36, 300, 240, 186, 147, 120, 99, 82, 72, 0, 329, 2730, 2310, 1920, 1605, 1356, 1155, 988, 848, 730, 658, 0, 3655, 30240, 26460, 22890, 19845, 17280, 15105, 13242, 11634, 10240, 9027, 7968, 7310, 0, 47844
Offset: 0
The first few rows of T(n,k) are:
0, 0;
2, 0, 0, 1;
8, 4, 2, 2, 0, 5;
42, 30, 20, 15, 12, 10, 0, 36;
For n = 2, let the four vertices be A, B, C, D. The diagram consisting of the chords (A,B) and (C,D) has no bubbles. The diagram consisting of the chords (A,D) and (B,C) has two bubbles of size 1: The vertex A is one bubble and the vertex D is the other. The diagram consisting of the chords (A,C) and (B,D) is itself a bubble of size 4. Hence T(2,1) = 2 and T(2,4) = 1.
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N=2*n;
G=0; for(j=0,j=N/2, G=G+taylor((1/((1 + w*(-1 + w*y^2))^2))*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*(2*j)!/j!* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j/j!*w*y^2*((-2 + 2*w + (3 -4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6 )*(2*j)!+(-y^4 + w*y^4+ w*y^6 - 2*w^2*y^6 + w^3*y^8 )*(2*j+2)!)),y,N+1); );
Tn=vector(N,x,0);
for(k=1,k=N,Tn[k]=polcoeff(polcoeff(G,N,y),k,w););
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