cp's OEIS Frontend

From Petros Hadjicostas, Jul 07 2018: (Start)

Column k of this array is the "BIK" (reversible, indistinct, unlabeled) transform of k,0,0,0,....

Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BIK(c_k(n): n >= 1) be the output sequence under Bower's BIK transform. It can proved that the g.f. of BIK(c_k(n): n >= 1) is A_k(x) = (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))). (See the comments for sequence A001224.)

For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k is (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))) = (1/2)*(k*x/(1-k*x) + (k*x^2 + k*x)/(1-k*x^2)) = (2 + (1-k)*x - 2*k*x^2)*k*x/(2*(1-k*x^2)*(1-k*x)).

Using the first form the g.f. above and the expansion 1/(1-y) = 1 + y + y^2 + ..., we can easily prove J.-F. Alcover's formula T(n,k) = (k^n + k^((n + mod(n,2))/2))/2.

(End)

The wire stays in the plane, there are n bends, each is R,L or O; turning the wire over does not count as a new figure.

Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, with n "corners"; the symmetry group is S4, reversing a walk does not count as different. Simply interpret R,L,O as instructions to turn R, turn L, or retrace the last step. Walks are not self-avoiding.

Also, it appears that a(n) gives the number of equivalence classes of n-tuples of 0, 1 and 2, where two n-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes taking the 2's complement (C(x)=2-x). This has been verified up to a(19)=290585050. Example: for n=3 there are ten equivalence classes {000, 222}, {001, 100, 122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, so a(3)=10. - John W. Layman, Oct 13 2009

There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013

If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017

a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018

From Allan Bickle, Jun 02 2022: (Start)

a(n) is the number of (unlabeled) 3-paths with n+6 vertices. (A 3-path with order n at least 5 can be constructed from a 4-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to an existing 3-clique containing an existing 3-leaf.)

Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

a(n) is also the number of distinct planar embeddings of the (n+1)-alkane graph (up to at least n=9, and likely for all n). - Eric W. Weisstein, May 21 2024

Number of bracelet structures of length n using exactly k different colored beads. Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure. - Andrew Howroyd, Apr 06 2017

The number of achiral structures (A) is given in A140735 (odd n) and A293181 (even n). The number of achiral structures plus twice the number of chiral pairs (A+2C) is given in A152175. These can be used to determine A+C by taking half their average, as is done in the Mathematica program. - Robert A. Russell, Feb 24 2018

T(n,k)=pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

Number of n-bead necklace structures using exactly k different colored beads. Turning over the necklace is not allowed. Permuting the colors does not change the structure. - Andrew Howroyd, Apr 06 2017

From David W. Wilson, Mar 10 2005: (Start)

Let M(n) be a map of n regions in a row. The number of ways to color M(n) if same-color regions are allowed to touch is given by A000110(n).

For example, M(4) has A000110(4) = 15 such colorings: aaaa aaab aaba aabb aabc abaa abab abac abba abbb abbc abca abcb abcc abcd.

The number of colorings of M(n) that are equivalent to their reverse is given by A080107(n). For example, M(4) has A080107(4) = 7 colorings that are equivalent to their reversal: aaaa aabb abab abba abbc abca abcd.

The number of distinct colorings when reversals are counted as equivalent is given by (A000110(n) + A080107(n))/2, which is essentially the present sequence. M(4) has 11 colorings that are distinct up to reversal: aaaa aaab aaba aabb aabc abab abac abba abbc abca abcd.

We can redo the whole analysis, this time forbidding same-color regions to touch. When we do, we get the same sequences, each with an extra 1 at the beginning. (End)

Note that A056325 gives the number of reversible string structures with n beads using a maximum of six different colors ... and, of course, any limit on the number of colors will be the same as this sequence above up to that number.

If the two ends of the line are distinguishable, so that 'abcb' and 'abac' are distinct, we get the Bell numbers, A000110(n - 1).

With a different offset, number of set partitions of [n] up to reflection (i<->n+1-i). E.g., there are 4 partitions of [3]: 123, 1-23, 13-2, 1-2-3 but not 12-3 because it is the reflection of 1-23. - David Callan, Oct 10 2005

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Two color patterns are equivalent if the colors are permuted.

In an unoriented row, chiral pairs are counted as one.

T(n,k) = Pi_k(P_n) which is the number of non-equivalent partitions of the path on n vertices, with at most k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

From Allan Bickle, Apr 05 2022: (Start)

The columns count unlabeled k-paths with n+k+2 vertices. (A k-path with order n at least k+2 is a k-tree with exactly two k-leaves (vertices of degree k). It can be constructed from a clique with k+1 vertices by iteratively adding a new degree k vertex adjacent to an existing clique containing an existing k-leaf.)

Recurrences for the columns appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly three different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

If the top entry of the triangle is changed from 0 to 1, this is the number of non-equivalent distinguishing partitions of the path on n vertices (n >= 1) with exactly k parts (1 <= k <= n). - Bahman Ahmadi, Aug 21 2019