A303732
Number of noncrossing path sets on n nodes up to rotation with each path having a prime number of nodes.
Original entry on oeis.org
1, 0, 1, 1, 1, 7, 8, 45, 96, 258, 1260, 2511, 12594, 35799, 126043, 482640, 1487929, 6012740, 20051360, 74529198, 276148256, 977824914, 3744986184, 13527623583, 50997301218, 190934525258, 711190503929, 2707743977818, 10155615925523, 38691707792278
Offset: 0
-
\\ number of path sets with restricted path lengths
NCPathSetsModCyclic(v)={ my(n=#v);
my(p=serreverse(x/(1 + x*v[1] + sum(k=2, #v, (k*2^(k-3))*x^k*v[k])) + O(x^2*x^n) )/x);
my(vars=variables(p));
my(h=substvec(p + O(x^(n\2+1)),vars,apply(t->t^2, vars)));
my(q=x*deriv(p)/p);
my(Q=sum(i=1, #v\2, v[2*i]*2^(i-1)*(x^2*h)^i));
1 + Q/2 + intformal((p - 1 + sum(d=2, n, eulerphi(d)*substvec(q + O(x^(n\d+1)), vars, apply(t->t^d, vars))))/x)
}
Vec(NCPathSetsModCyclic(vector(30, k, isprime(k))))
A303839
Number of noncrossing path sets on n nodes up to rotation and reflection with each path having at least two nodes.
Original entry on oeis.org
1, 0, 1, 1, 3, 5, 17, 40, 138, 430, 1546, 5478, 20525, 77310, 298301, 1161692, 4583525, 18239037, 73221198, 296046399, 1205038270, 4933969005, 20311807087, 84029440358, 349201537324, 1457205298510, 6104204225832, 25661191956781, 108231773165825
Offset: 0
Case n=4: There are 3 possibilities:
.
o---o o o o---o
| | /
o---o o---o o---o
.
A303729
Number of noncrossing path sets on n nodes with each path having a prime number of nodes.
Original entry on oeis.org
1, 0, 1, 3, 2, 35, 32, 315, 746, 2304, 12422, 27621, 150729, 465387, 1762427, 7239244, 23799382, 102216580, 360900542, 1416054762, 5522838696, 20534319262, 82389314900, 311135342409, 1223933415631, 4773363130810, 18490946264039, 73109087367264, 284357219601461
Offset: 0
-
seq[n_] := InverseSeries[x/(1 + Sum[If[PrimeQ[k], k*2^(k-3)*x^k, 0], {k, 2, n}]) + O[x]^(n+2), x]/x;
CoefficientList[seq[28], x] (* Jean-François Alcover, May 15 2018, translated from PARI *)
-
seq(n)={Vec(serreverse(x/(1 + sum(k=2, n, if(isprime(k), k*2^(k-3)*x^k))) + O(x^(n+2)) )/x)}
A303835
Number of noncrossing path sets on n nodes up to rotation and reflection with isolated vertices allowed.
Original entry on oeis.org
1, 1, 2, 3, 8, 19, 64, 212, 833, 3360, 14476, 63848, 289892, 1338000, 6275589, 29791100, 142973014, 692507861, 3382070233, 16638445745, 82395500651, 410463736691, 2055858519575, 10347925039015, 52321093290715, 265648012207312, 1353953547877556, 6925400869302520
Offset: 0
Case n=3: There are 3 possibilities:
.
o o o
/ \
o o o---o o o
.
Case n=4: There are 8 possibilities:
.
o o o o o o o o o o o---o o o o---o
/ | / | | /
o o o---o o o o---o o---o o---o o---o o---o
.
A303868
Triangle read by rows: T(n,k) = number of noncrossing path sets on n nodes up to rotation and reflection with k paths and isolated vertices allowed.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 7, 6, 2, 1, 6, 20, 23, 11, 3, 1, 10, 50, 80, 51, 17, 3, 1, 20, 136, 285, 252, 109, 26, 4, 1, 36, 346, 966, 1119, 652, 200, 36, 4, 1, 72, 901, 3188, 4782, 3623, 1502, 352, 50, 5, 1, 136, 2264, 10133, 19116, 18489, 9949, 3120, 570, 65, 5, 1
Offset: 1
Triangle begins:
1;
1, 1;
1, 1, 1;
2, 3, 2, 1;
3, 7, 6, 2, 1;
6, 20, 23, 11, 3, 1;
10, 50, 80, 51, 17, 3, 1;
20, 136, 285, 252, 109, 26, 4, 1;
36, 346, 966, 1119, 652, 200, 36, 4, 1;
72, 901, 3188, 4782, 3623, 1502, 352, 50, 5, 1;
...
-
\\ See A303731 for NCPathSetsModDihedral
{ my(rows=Vec(NCPathSetsModDihedral(vector(10, k, y))-1));
for(n=1, #rows, for(k=1, n, print1(polcoeff(rows[n],k), ", ")); print;) }
Showing 1-5 of 5 results.