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 1-10 of 17 results. Next

A082315 Permutation A057501 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 9, 11, 14, 16, 19, 21, 22, 18, 17, 20, 13, 12, 10, 15, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 58, 59, 62, 63, 64, 49, 50, 46, 45, 48, 55, 54, 57, 61, 35, 36, 32, 31, 34, 27, 26, 24, 29, 41, 40, 38, 43, 52, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003, Proposed by Wouter Meeussen

Keywords

Links

Crossrefs

Inverse of A082316. a(n) = A082313(A057164(n)). Occurs in A073200 as row 34359740687. Cf. also A082317-A082324.
Number of cycles: A054357. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057501(A057501(n)).

A085159 Permutation of natural numbers induced by the Catalan bijection gma085159 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 13, 21, 15, 18, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 27, 35, 29, 32, 36, 58, 62, 40, 38, 41, 49, 43, 46, 50, 63, 52, 55, 59, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085160. a(n) = A085161(A085160(A085161(n))) = A085169(A082315(A085170(n))) = A074684(A082315(A074683(n))) = A085173(A085173(n)). Occurs in A073200. Cf. also A085165-A085168, A086429. Scheme-function app-to-xrt given in A085203.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085160 Permutation of natural numbers induced by the Catalan bijection gma085160 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 18, 11, 20, 12, 14, 21, 13, 15, 19, 22, 23, 45, 24, 44, 46, 25, 48, 26, 42, 49, 27, 43, 47, 50, 28, 54, 29, 53, 55, 30, 57, 31, 37, 58, 32, 38, 56, 59, 33, 61, 34, 39, 62, 35, 40, 51, 63, 36, 41, 52, 60, 64, 65, 129, 66, 128, 130
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085159. a(n) = A085161(A085159(A085161(n))) = A085169(A082316(A085170(n))) = A074684(A082316(A074683(n))) = A085174(A085174(n)). Occurs in A073200. Cf. also A085165-A085168, A086430.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A303912 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled k-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 6, 1, 1, 1, 5, 10, 19, 10, 1, 1, 1, 6, 15, 44, 57, 28, 1, 1, 1, 7, 21, 85, 197, 258, 63, 1, 1, 1, 8, 28, 146, 510, 1228, 1110, 190, 1, 1, 1, 9, 36, 231, 1101, 4051, 7692, 5475, 546, 1, 1, 1, 10, 45, 344, 2100, 10632, 33130, 52828, 27429, 1708, 1
Offset: 0

Views

Author

Andrew Howroyd, May 02 2018

Keywords

Comments

A k-ary cactus is a planar k-gonal cactus with vertices on each polygon numbered 1..k counterclockwise with shared vertices having the same number. In total there are always exactly k ways to number a given cactus since all polygons are connected. See the reference for a precise definition.

Examples

			Array begins:
===============================================================
n\k| 1   2     3      4       5        6        7         8
---+-----------------------------------------------------------
0  | 1   1     1      1       1        1        1         1 ...
1  | 1   1     1      1       1        1        1         1 ...
2  | 1   2     3      4       5        6        7         8 ...
3  | 1   3     6     10      15       21       28        36 ...
4  | 1   6    19     44      85      146      231       344 ...
5  | 1  10    57    197     510     1101     2100      3662 ...
6  | 1  28   258   1228    4051    10632    23884     47944 ...
7  | 1  63  1110   7692   33130   107062   285390    662628 ...
8  | 1 190  5475  52828  291925  1151802  3626295   9711032 ...
9  | 1 546 27429 373636 2661255 12845442 47813815 147766089 ...
...

Links

Crossrefs

Columns 2..7 are A054357, A052393, A052394, A054363, A054366, A054369.
Cf. A070914, A303694, A303913.

Programs

  • Mathematica
    T[0, _] = 1;
T[n_, k_] := DivisorSum[n, EulerPhi[n/#] Binomial[k #, #]&]/n - (k-1) Binomial[n k, n]/((k-1) n + 1);
Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 22 2018 *)
  • PARI
    T(n,k)={if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1))}

Formula

T(n,k) = (Sum_{d|n} phi(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1) for n > 0.
T(n,k) ~ A070914(n,k-1)/n for fixed k > 1.

A086430 Permutation of natural numbers induced by the Catalan bijection gma086430 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 18, 12, 10, 15, 20, 21, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 45, 46, 48, 49, 50, 31, 32, 26, 24, 29, 40, 38, 43, 52, 54, 55, 57, 58, 59, 34, 35, 27, 41, 61, 62, 63, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086429. a(n) = A086428(A086428(n)) = A086431(A086429(A086431(n))) = A057164(A085160(A057164(n))) = A086425(A082316(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

Original entry on oeis.org

1, 2, 3, 6, 10, 24, 49, 130, 336, 980, 2904, 9176, 29432, 97356, 326399, 1111770, 3825238, 13293456, 46553116, 164200028, 582706692, 2079517924, 7458493728, 26874412064, 97241528200, 353223728624, 1287668381250, 4709805627484
Offset: 1

Views

Author

David Callan and Len Smiley, Oct 21 2005

Keywords

Comments

These may be viewed as bracelets (able to be turned over in space) designed with n beads on a circle, each of which is a vertex of exactly one of a set of non-touching internal polygons (which may be 1-gons (beads), 2-gons (2 connected beads), etc.).

References

  • S.-C. Chang, J. L. Jacobsen, J. Salas, R. Shrock, "Exact Potts model partition functions for strips of the triangular lattice", J. Statist. Phys. 114, nos.3-4, pp. 763-823 [Corollary 2.1]
  • Motzkin, T. "Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-Associative Products." Bull. Amer. Math. Soc. 54, page 360, 1948.

Links

Crossrefs

Cf. A209612.

Programs

  • Mathematica
    Table[Length[EquivalenceClasses[NCPartitions[n], groupDihedral[n]]], {n, 9}]

Formula

(A054357(n) + A001405(n))/2.

A082316 Permutation A057502 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 5, 7, 6, 9, 21, 10, 20, 19, 11, 22, 12, 17, 16, 13, 18, 14, 15, 23, 58, 24, 57, 56, 25, 59, 26, 54, 53, 27, 55, 51, 52, 28, 62, 29, 61, 60, 30, 63, 31, 45, 44, 32, 46, 42, 43, 33, 64, 34, 48, 47, 35, 49, 37, 38, 36, 50, 39, 40, 41, 65, 170, 66, 169, 168
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003, Proposed by Wouter Meeussen

Keywords

Links

Crossrefs

Inverse of A082315. a(n) = A057164(A082313(n)). Occurs in A073200 as row 549755978251. Cf. also A082317-A082324.
Number of cycles: A054357. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057502(A057502(n))

A086429 Permutation of natural numbers induced by the Catalan bijection gma086429 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 21, 11, 18, 12, 14, 15, 13, 19, 20, 22, 23, 45, 24, 44, 58, 25, 46, 26, 42, 43, 27, 56, 57, 63, 28, 48, 29, 47, 59, 30, 49, 31, 37, 38, 32, 39, 40, 41, 33, 50, 34, 51, 52, 35, 53, 54, 55, 36, 60, 61, 62, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086430. a(n) = A086427(A086427(n)) = A086431(A086430(A086431(n))) = A057164(A085159(A057164(n))) = A086425(A082315(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A209805 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 10, 10, 3, 1, 1, 3, 15, 25, 15, 3, 1, 1, 4, 26, 64, 64, 26, 4, 1, 1, 4, 38, 132, 196, 132, 38, 4, 1, 1, 5, 56, 256, 536, 536, 256, 56, 5, 1, 1, 5, 75, 450, 1260, 1764, 1260, 450, 75, 5, 1
Offset: 1

Views

Author

Tilman Piesk, Mar 13 2012

Keywords

Comments

Like the Narayana triangle A001263 (and unlike A152175) this triangle is symmetric.
The diagonal entries are 1, 1, 4, 25, 196, 1764, ... which is probably sequence A001246 - the squares of the Catalan numbers.
The above conjecture about the diagonal entries T(2*n-1, n) is true since gcd(2*n-1, n) = gcd(2*n-1, n-1) = 1 and then T(2*n-1, n) simplifies to A001246(n-1) using the formula given below. - Andrew Howroyd, Nov 15 2017

Examples

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   2,   2,   1;
  1,   2,   4,   2,   1;
  1,   3,  10,  10,   3,   1;
  1,   3,  15,  25,  15,   3,   1;
  1,   4,  26,  64,  64,  26,   4,   1;
  1,   4,  38, 132, 196, 132,  38,   4,   1;
  1,   5,  56, 256, 536, 536, 256,  56,   5,   1;

Links

Crossrefs

Cf. A054357 (row sums), A001246 (square Catalan numbers).
Cf. A001263, A103371, A132812, A209612.

Programs

  • Mathematica
    b[n_, k_] := Binomial[n-1, n-k] Binomial[n, n-k];
T[n_, k_] := (DivisorSum[GCD[n, k], EulerPhi[#] b[n/#, k/#]&] + DivisorSum[GCD[n, k - 1], EulerPhi[#] b[n/#, (n + 1 - k)/#]&] - k Binomial[n, k]^2/(n - k + 1))/n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)
  • PARI
    b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);
T(n,k)=(sumdiv(gcd(n,k), d, eulerphi(d)*b(n/d,k/d)) + sumdiv(gcd(n,k-1), d, eulerphi(d)*b(n/d,(n+1-k)/d)) - k*binomial(n,k)^2/(n-k+1))/n; \\ Andrew Howroyd, Nov 15 2017

Formula

T(n,k) = (1/n)*((Sum_{d|gcd(n,k)} phi(d)*A103371(n/d-1,k/d-1)) + (Sum_{d|gcd(n,k-1)} phi(d)*A103371(n/d-1,(n+1-k)/d-1)) - A132812(n,k)). - Andrew Howroyd, Nov 15 2017

A085165 A057163-conjugate of A085159.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 12, 16, 17, 21, 10, 18, 11, 14, 15, 13, 19, 20, 22, 23, 26, 30, 31, 35, 42, 43, 44, 45, 49, 56, 57, 58, 63, 24, 32, 47, 46, 59, 25, 48, 28, 37, 38, 29, 39, 40, 41, 27, 50, 33, 51, 52, 34, 53, 54, 55, 36, 60, 61, 62, 64, 65, 68, 72, 73, 77
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085166. a(n) = A057163(A085159(A057163(n))) = A085162(A085166(A085162(n))). Occurs in A073200. Cf. also A085162, A086429, A086430.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).
Showing 1-10 of 17 results. Next

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