A303731
Number of noncrossing path sets on n nodes up to rotation and reflection with each path having a prime number of nodes.
Original entry on oeis.org
1, 0, 1, 1, 1, 5, 6, 27, 53, 140, 649, 1297, 6355, 18038, 63226, 241741, 744711, 3008107, 10028056, 37270169, 138083464, 488933323, 1872525356, 6763888465, 25498771059, 95467533318, 355595703773, 1353873044078, 5077809606803, 19345857682140, 73533468653115
Offset: 0
-
\\ number of path sets with restricted path lengths
NCPathSetsModDihedral(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(R=v[1]*x + sum(i=1, (#v-1)\2, v[2*i+1]*2^(i-1)*x*(x^2*h)^i), Q=sum(i=1, #v\2, v[2*i]*2^(i-1)*(x^2*h)^i), T=intformal((p - 1 + sum(d=2,n, eulerphi(d)*substvec(q + O(x^(n\d+1)), vars, apply(t->t^d, vars))))/x));
O(x*x^n) + (1 + T + (1 + Q + (1+R)^2*h/(1-Q) + v[2]*x^2*h)/2)/2;
}
Vec(NCPathSetsModDihedral(vector(30, k, isprime(k))))
A303844
Number of noncrossing path sets on n nodes up to rotation with each path having at least two nodes.
Original entry on oeis.org
1, 0, 1, 1, 4, 7, 26, 69, 246, 818, 2976, 10791, 40591, 153959, 594753, 2320696, 9159498, 36467012, 146411208, 592046830, 2409946566, 9867745442, 40623068380, 168058068487, 698400767839, 2914407151002, 12208398647345, 51322369218674, 216463504458521
Offset: 0
Case n=4: There are 4 possibilities:
.
o---o o o 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)}
A303869
Triangle read by rows: T(n,k) = number of noncrossing path sets on n nodes up to rotation with k paths and isolated vertices allowed.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 3, 4, 2, 1, 4, 11, 8, 2, 1, 10, 34, 39, 16, 3, 1, 16, 92, 144, 90, 25, 3, 1, 36, 256, 545, 473, 197, 40, 4, 1, 64, 672, 1878, 2184, 1246, 370, 56, 4, 1, 136, 1762, 6296, 9436, 7130, 2910, 658, 80, 5, 1, 256, 4480, 20100, 38025, 36690, 19698, 6090, 1080, 105, 5, 1
Offset: 1
Triangle begins:
1;
1, 1;
1, 1, 1;
3, 4, 2, 1;
4, 11, 8, 2, 1;
10, 34, 39, 16, 3, 1;
16, 92, 144, 90, 25, 3, 1;
36, 256, 545, 473, 197, 40, 4, 1;
64, 672, 1878, 2184, 1246, 370, 56, 4, 1;
136, 1762, 6296, 9436, 7130, 2910, 658, 80, 5, 1;
...
-
\\ See A303732 for NCPathSetsModCyclic
{ my(rows=Vec(NCPathSetsModCyclic(vector(10, k, y))-1));
for(n=1, #rows, for(k=1,n,print1(polcoeff(rows[n],k), ", ")); print;)}
A303836
Number of noncrossing path sets on n nodes up to rotation with isolated vertices allowed.
Original entry on oeis.org
1, 1, 2, 3, 10, 26, 103, 371, 1552, 6475, 28414, 126530, 577188, 2670332, 12538434, 59554199, 285882600, 1384875627, 6763821250, 33276183371, 164789380052, 820923863918, 4111708742153, 20695831549310, 104642143845428, 531295928725508, 2707906874407464
Offset: 0
Case n=4: There are 10 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 o---o o---o o---o o---o
.
Showing 1-5 of 5 results.