A211353
Refined triangle A211357: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 4, 6, 3, 5, 1, 2, 1, 1, 1, 1, 3, 5, 10, 5, 15, 3, 5, 6, 3, 1, 3, 1, 1, 1, 1, 4, 7, 19, 10, 35, 7, 19, 21, 12, 4, 21, 7, 7, 1, 3, 4, 4, 1, 1, 1, 1, 1, 4, 10, 28, 14, 70, 14, 48, 56, 28, 10
Offset: 1
A211359
Triangle read by rows: T(n,k) is the number of noncrossing partitions up to rotation and reflection of an n-set that contain k singleton blocks.
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 3, 2, 2, 0, 1, 5, 4, 8, 3, 3, 0, 1, 6, 11, 12, 12, 4, 3, 0, 1, 14, 21, 39, 24, 22, 5, 4, 0, 1, 22, 55, 84, 85, 48, 30, 7, 4, 0, 1, 51, 124, 245, 228, 190, 82, 46, 8, 5, 0, 1, 95, 327, 620, 730, 570, 350, 136, 60
Offset: 0
From _Andrew Howroyd_, May 02 2018: (Start)
Triangle begins:
1;
0, 1;
1, 0, 1;
1, 1, 0, 1;
2, 1, 2, 0, 1;
2, 3, 2, 2, 0, 1;
5, 4, 8, 3, 3, 0, 1;
6, 11, 12, 12, 4, 3, 0, 1;
14, 21, 39, 24, 22, 5, 4, 0, 1;
22, 55, 84, 85, 48, 30, 7, 4, 0, 1;
51, 124, 245, 228, 190, 82, 46, 8, 5, 0, 1;
...
(End)
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\\ See A303875 for NCPartitionsModDihedral
{ my(rows=Vec(NCPartitionsModDihedral(vector(10, k, if(k==1,y,1)))));
for(n=1, #rows, for(k=0, n-1, print1(polcoeff(rows[n], k), ", ")); print; ) } \\ Andrew Howroyd, May 02 2018
A295198
Number of noncrossing partitions up to rotation of an n-set without singleton blocks.
Original entry on oeis.org
1, 0, 1, 1, 2, 2, 5, 6, 15, 28, 67, 145, 368, 870, 2211, 5549, 14290, 36824, 96347, 252927, 670142, 1783770, 4777951, 12855392, 34756783, 94345664, 257114389, 703150507, 1929404736, 5310364234, 14658134277, 40569137070, 112566363319, 313074271844, 872677323283
Offset: 0
Cf.
A005043 (noncrossing partitions of an n-set without singleton blocks).
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b[0] = 1; b[1] = 0; b[n_] := b[n] = (n-1)*(2*b[n-1] + 3*b[n-2])/(n+1);
a[0] = 1; a[n_] := (b[n] + Sum[EulerPhi[n/d]*Coefficient[(1 + x + x^2)^d, x, d], {d, Most @ Divisors[n]}])/n;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)
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\\ here b(n) is A005043.
b(n) = {polcoeff(serreverse((x - x^3) / (1 + x^3) + x * O(x*x^n)), n+1)}
a(n) = {if(n<1, n==0, (b(n) + sumdiv(n,d, if(d
Showing 1-3 of 3 results.
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