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A320751
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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).
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7
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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
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OFFSET
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1,14
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COMMENTS
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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
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LINKS
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FORMULA
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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) = Sum_{j=1..k} A320525(n,j).
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EXAMPLE
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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.
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MATHEMATICA
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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
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CROSSREFS
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As k increases, columns converge to A320937.
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KEYWORD
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AUTHOR
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STATUS
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approved
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