A303875
Number of noncrossing partitions of an n-set up to rotation and reflection with all blocks having a prime number of elements.
Original entry on oeis.org
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
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\\ 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))))
A210735
Number of Dyck n-paths all of whose ascents and descents have prime lengths.
Original entry on oeis.org
1, 0, 1, 1, 1, 4, 2, 10, 10, 22, 46, 64, 167, 245, 560, 1035, 1978, 4210, 7715, 16497, 31929, 65216, 133295, 266244, 553750, 1116404, 2308931, 4738660, 9742795, 20204902, 41622910, 86539105, 179358694, 373018581, 777157221, 1618773690, 3382590684, 7065505631
Offset: 0
a(0) = 1: the empty path.
a(1) = 0.
a(2) = 1: UUDD.
a(3) = 1: UUUDDD.
a(4) = 1: UUDDUUDD.
a(5) = 4: UUDDUUUDDD, UUUDDDUUDD, UUUDDUUDDD, UUUUUDDDDD.
a(6) = 2: UUDDUUDDUUDD, UUUDDDUUUDDD.
a(7) = 10: UUDDUUDDUUUDDD, UUDDUUUDDDUUDD, UUDDUUUDDUUDDD, UUDDUUUUUDDDDD, UUUDDDUUDDUUDD, UUUDDUUDDDUUDD, UUUDDUUDDUUDDD, UUUUUDDDDDUUDD, UUUUUDDUUDDDDD, UUUUUUUDDDDDDD.
a(8) = 10: UUDDUUDDUUDDUUDD, UUDDUUUDDDUUUDDD, UUUDDDUUDDUUUDDD, UUUDDDUUUDDDUUDD, UUUDDDUUUDDUUDDD, UUUDDDUUUUUDDDDD, UUUDDUUDDDUUUDDD, UUUDDUUUDDDUUDDD, UUUUUDDDDDUUUDDD, UUUUUDDDUUUDDDDD.
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with(numtheory):
b:= proc(x, y, u) option remember;
`if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);
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b[x_, y_, u_] := b[x, y, u] = If[x<0 || yJean-François Alcover, Mar 24 2017, translated from Maple *)
A303874
Number of noncrossing partitions of an n-set up to rotation with all blocks having a prime number of elements.
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 3, 5, 8, 17, 37, 71, 179, 366, 919, 2069, 5027, 12053, 29098, 71846, 175485, 437438, 1087122, 2723326, 6860525, 17301606, 43957596, 111748571, 285591775, 731432424, 1879009622, 4841510973, 12500324496, 32366232373, 83962263464, 218309244314
Offset: 0
-
\\ number of partitions with restricted block sizes
NCPartitionsModCyclic(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);
my(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
}
Vec(NCPartitionsModCyclic(vector(40, k, isprime(k))))
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