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A320751 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a row of length n using k or fewer colors (subsets). 7
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 4, 16, 12, 0, 0, 0, 1, 4, 20, 52, 28, 0, 0, 0, 1, 4, 20, 80, 169, 56, 0, 0, 0, 1, 4, 20, 86, 336, 520, 120, 0, 0, 0, 1, 4, 20, 86, 400, 1344, 1600, 240, 0, 0, 0, 1, 4, 20, 86, 409, 1852, 5440, 4840, 496, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,14

COMMENTS

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

T(n,k)=Xi_k(P_n) which is the number of non-equivalent distinguishing 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. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. - Bahman Ahmadi, Sep 02 2019

LINKS

Table of n, a(n) for n=1..78.

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

FORMULA

T(n,k) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where S2 is the Stirling subset number A008277 and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).

T(n,k) = (A278984(k,n) - A305749(n,k)) / 2 = A278984(k,n) - A320750(n,k) = A320750(n,k) - A305749(n,k).

T(n,k) = Sum_{j=1..k} A320525(n,j).

EXAMPLE

Array begins with T(1,1):

0   0     0      0       0       0       0       0       0       0 ...

0   0     0      0       0       0       0       0       0       0 ...

0   1     1      1       1       1       1       1       1       1 ...

0   2     4      4       4       4       4       4       4       4 ...

0   6    16     20      20      20      20      20      20      20 ...

0  12    52     80      86      86      86      86      86      86 ...

0  28   169    336     400     409     409     409     409     409 ...

0  56   520   1344    1852    1976    1988    1988    1988    1988 ...

0 120  1600   5440    8868   10168   10388   10404   10404   10404 ...

0 240  4840  21760   42892   54208   57108   57468   57488   57488 ...

0 496 14641  87296  210346  299859  331705  337595  338155  338180 ...

0 992 44044 349184 1038034 1699012 2012202 2091458 2102518 2103348 ...

For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA.

For T(4,3)=4, the above, AABC-ABCC, and ABAC-ABCB.

MATHEMATICA

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)

Table[Sum[StirlingS2[n, j] - Ach[n, j], {j, k-n+1}]/2, {k, 15}, {n, k}] // Flatten

CROSSREFS

Columns 1-6 are A000004, A122746(n-3), A107767(n-1), A320934, A320935, A320936.

As k increases, columns converge to A320937.

Cf. transpose of A278984 (oriented), A320750 (unoriented), A305749 (achiral).

Partial column sums of A320525.

Sequence in context: A187080 A301342 A226369 * A263764 A325668 A070202

Adjacent sequences:  A320748 A320749 A320750 * A320752 A320753 A320754

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, Oct 27 2018

STATUS

approved

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Last modified September 19 11:23 EDT 2021. Contains 347556 sequences. (Running on oeis4.)