A303875 Number of noncrossing partitions of an n-set up to rotation and reflection with all blocks having a prime number of elements.

1, 0, 1, 1, 1, 2, 3, 5, 7, 14, 26, 49, 107, 215, 502, 1112, 2619, 6220, 14807, 36396, 88397, 219920, 545196, 1364669, 3434436, 8658463, 21989434, 55893852, 142823174, 365766327, 939575265, 2420885031, 6250344302, 16183450744, 41981605437, 109155492638

Offset: 0

Author

Andrew Howroyd, May 01 2018

Keywords

• nonn

Comments

The number of such noncrossing partitions counted distinctly is given by A210737.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A111275, A185100 (1 or 2), A210737, A303874, A303931.

Programs

PARI

\\ number of partitions with restricted block sizes
NCPartitionsModDihedral(v)={ my(n=#v);
my(p=serreverse( x/(1 + sum(k=1, #v, x^k*v[k])) + O(x^2*x^n) )/x);
my(vars=variables(p));
my(varpow(r,d)=substvec(r + O(x^(n\d+1)), vars, apply(t->t^d, vars)));
my(q=x*deriv(p)/p, h=varpow(p,2));
my(R=sum(i=0, (#v-1)\2, v[2*i+1]*x*(x^2*h)^i), Q=sum(i=1, #v\2, v[2*i]*(x^2*h)^i), T=sum(k=1, #v, my(t=v[k]); if(t, x^k*t*sumdiv(k, d, eulerphi(d) * varpow(p,d)^(k/d))/k)));
(T + 2 + intformal(sum(d=1, n, eulerphi(d)*varpow(q,d))/x) - p + (1 + Q + (1+R)^2*h/(1-Q))/2)/2 + O(x*x^n)
}
Vec(NCPartitionsModDihedral(vector(40,k,isprime(k))))