%I #20 Nov 05 2019 06:00:02
%S 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,
%T 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,
%U 20,86,400,1344,1600,240,0,0,0,1,4,20,86,409,1852,5440,4840,496,0
%N 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).
%C Two color patterns are equivalent if the colors are permuted.
%C A chiral row is not equivalent to its reverse.
%C 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
%H B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, <a href="https://arxiv.org/abs/1910.12102">Number of Distinguishing Colorings and Partitions</a>, arXiv:1910.12102 [math.CO], 2019.
%F 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)).
%F T(n,k) = (A278984(k,n) - A305749(n,k)) / 2 = A278984(k,n) - A320750(n,k) = A320750(n,k) - A305749(n,k).
%F T(n,k) = Sum_{j=1..k} A320525(n,j).
%e Array begins with T(1,1):
%e 0 0 0 0 0 0 0 0 0 0 ...
%e 0 0 0 0 0 0 0 0 0 0 ...
%e 0 1 1 1 1 1 1 1 1 1 ...
%e 0 2 4 4 4 4 4 4 4 4 ...
%e 0 6 16 20 20 20 20 20 20 20 ...
%e 0 12 52 80 86 86 86 86 86 86 ...
%e 0 28 169 336 400 409 409 409 409 409 ...
%e 0 56 520 1344 1852 1976 1988 1988 1988 1988 ...
%e 0 120 1600 5440 8868 10168 10388 10404 10404 10404 ...
%e 0 240 4840 21760 42892 54208 57108 57468 57488 57488 ...
%e 0 496 14641 87296 210346 299859 331705 337595 338155 338180 ...
%e 0 992 44044 349184 1038034 1699012 2012202 2091458 2102518 2103348 ...
%e For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA.
%e For T(4,3)=4, the above, AABC-ABCC, and ABAC-ABCB.
%t 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 *)
%t Table[Sum[StirlingS2[n,j] - Ach[n,j], {j,k-n+1}]/2, {k,15}, {n,k}] // Flatten
%Y Columns 1-6 are A000004, A122746(n-3), A107767(n-1), A320934, A320935, A320936.
%Y As k increases, columns converge to A320937.
%Y Cf. transpose of A278984 (oriented), A320750 (unoriented), A305749 (achiral).
%Y Partial column sums of A320525.
%K nonn,tabl,easy
%O 1,14
%A _Robert A. Russell_, Oct 27 2018