A284949 Triangle read by rows: T(n,k) = number of reversible string structures of length n using exactly k different symbols.

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 9, 15, 6, 1, 1, 19, 50, 37, 9, 1, 1, 35, 160, 183, 76, 12, 1, 1, 71, 502, 877, 542, 142, 16, 1, 1, 135, 1545, 3930, 3523, 1346, 242, 20, 1, 1, 271, 4730, 17185, 21393, 11511, 2980, 390, 25, 1

Offset: 1

Author

Andrew Howroyd, Apr 06 2017

Keywords

• nonn
• tabl

Comments

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

Number of k-block partitions of an n-set up to reflection.

T(n,k) = pi_k(P_n) which is the number of non-equivalent partitions of the path 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

Examples

			Triangle begins:
1;
1,   1;
1,   2,    1;
1,   5,    4,     1;
1,   9,   15,     6,     1;
1,  19,   50,    37,     9,     1;
1,  35,  160,   183,    76,    12,    1;
1,  71,  502,   877,   542,   142,   16,   1;
1, 135, 1545,  3930,  3523,  1346,  242,  20,  1;
1, 271, 4730, 17185, 21393, 11511, 2980, 390, 25, 1;
		

References

• M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
• Mohammad Hadi Shekarriz, GAP Program

Crossrefs

Columns 2..6 are A056326, A056327, A056328, A056329, A056330.

Row sums are A103293.

Partial row sums include A005418, A001998(n-1), A056323, A056324, A056325.

Cf. A277504, A008277 (set partitions), A152175 (up to rotation), A152176 (up to rotation and reflection), A304972 (achiral patterns).

Programs

Mathematica

(* achiral color patterns for row of n colors containing k different colors *)
Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[0==n, 1, 0], 1, If[n>0, 1, 0],
   (* else *) _, If[OddQ[n],
   Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
   Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1] + 2^i Ach[n-2-2i, k-2]),
   {i, 0, n/2-1}]]]
Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 15}, {k, 1, n}] // Flatten
(* Robert A. Russell, Feb 10 2018 *)

PARI

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(ReversiblePerms(n), k); \\ Andrew Howroyd, Oct 14 2017

PARI

\\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n,n,i,k,i>=k)); for(i=3, n, for(k=2, n, M[i,k]=k*M[i-2,k] + M[i-2,k-1] + if(k>2, M[i-2,k-2]))); M}
T(n)={(matrix(n, n, i, k, stirling(i, k, 2)) + Ach(n))/2}
{ my(A=T(10)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Sep 18 2019