cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A303929 Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation and reflection composed of n blocks of size k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 6, 1, 1, 1, 1, 3, 8, 13, 12, 1, 1, 1, 1, 4, 11, 34, 49, 27, 1, 1, 1, 1, 4, 16, 60, 169, 201, 65, 1, 1, 1, 1, 5, 20, 109, 423, 1019, 940, 175, 1, 1, 1, 1, 5, 26, 167, 918, 3381, 6710, 4643, 490, 1
Offset: 0

Views

Author

Andrew Howroyd, May 02 2018

Keywords

Examples

			=================================================================
n\k| 1   2    3     4      5       6       7       8        9
---+-------------------------------------------------------------
0  | 1   1    1     1      1       1       1       1        1 ...
1  | 1   1    1     1      1       1       1       1        1 ...
2  | 1   1    1     1      1       1       1       1        1 ...
3  | 1   2    2     3      3       4       4       5        5 ...
4  | 1   3    5     8     11      16      20      26       32 ...
5  | 1   6   13    34     60     109     167     257      359 ...
6  | 1  12   49   169    423     918    1741    3051     4969 ...
7  | 1  27  201  1019   3381    9088   20569   41769    77427 ...
8  | 1  65  940  6710  29335   96315  259431  607696  1280045 ...
9  | 1 175 4643 47104 266703 1072187 3417520 9240444 22066742 ...
...

Links

Crossrefs

Columns 2..5 are A006082(n+1), A082938, A303870, A303871.
Cf. A111275, A209612, A211359, A303694, A303875.

Programs

  • Mathematica
    u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
Table[T[n - k, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)
  • PARI
    \\ here c(n,k) is A303694
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
c(n, k)={if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d)) + sumdiv(gcd(n-1, k), d, eulerphi(d)*binomial(n*k/d, (n-1)/d)))/(k*n) - binomial(k*n, n)/(n*(k-1)+1))}
T(n,k)={(1/2)*(c(n,k) + if(n==0, 1, if(k%2, if(n%2, 2*u(n\2,k,(k+1)/2), u(n/2,k,1) + u(n/2-1,k,k)), e(n,k) + if(n%2, u(n\2,k,k/2)))/2))}

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

Views

Author

Tilman Piesk, Apr 12 2012

Keywords

Examples

			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)

Links

Crossrefs

Column k=0 is A303931.
Row sums are A111275.
Cf. A209612, A211357, A303875.

Programs

  • PARI
    \\ 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

A303931 Number of noncrossing partitions up to rotation and reflection of an n-set without singleton blocks.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 6, 14, 22, 51, 95, 232, 498, 1239, 2953, 7520, 18920, 49235, 127917, 338094, 896060, 2397627, 6439730, 17403252, 47207620, 128628877, 351676075, 964909660, 2655474962, 7329668097, 20285420790, 56284927718, 156539620498, 436343744531
Offset: 0

Views

Author

Andrew Howroyd, May 02 2018

Keywords

Links

Crossrefs

Column k=0 of A211359.
Cf. A295198, A303875.

Programs

  • PARI
    \\ See A303875 for NCPartitionsModDihedral
Vec(NCPartitionsModDihedral(vector(40, k, k>1))) \\ Andrew Howroyd, May 02 2018

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

Views

Author

Andrew Howroyd, May 01 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A054357 (unrestricted), A175954 (1 or 2), A210737, A295198, A303875.

Programs

  • PARI
    \\ 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))))
Showing 1-4 of 4 results.

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