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A320525
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Triangle read by rows: T(n,k) = number of chiral pairs of color patterns (set partitions) in a row of length n using exactly k colors (subsets).
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8
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0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 6, 10, 4, 0, 0, 12, 40, 28, 6, 0, 0, 28, 141, 167, 64, 9, 0, 0, 56, 464, 824, 508, 124, 12, 0, 0, 120, 1480, 3840, 3428, 1300, 220, 16, 0, 0, 240, 4600, 16920, 21132, 11316, 2900, 360, 20, 0, 0, 496, 14145, 72655, 123050, 89513, 31846, 5890, 560, 25, 0, 0, 992, 43052, 305140, 688850, 660978, 313190, 79256, 11060, 830, 30, 0
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OFFSET
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1,8
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COMMENTS
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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
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LINKS
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FORMULA
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T(n,k) = (S2(n,k) - A(n,k))/2, where S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
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EXAMPLE
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Triangle begins with T(1,1):
0;
0, 0;
0, 1, 0;
0, 2, 2, 0;
0, 6, 10, 4, 0;
0, 12, 40, 28, 6, 0;
0, 28, 141, 167, 64, 9, 0;
0, 56, 464, 824, 508, 124, 12, 0;
0, 120, 1480, 3840, 3428, 1300, 220, 16, 0;
0, 240, 4600, 16920, 21132, 11316, 2900, 360, 20, 0;
0, 496, 14145, 72655, 123050, 89513, 31846, 5890, 560, 25, 0;
0, 992, 43052, 305140, 688850, 660978, 313190, 79256, 11060, 830, 30, 0;
...
For T(3,2)=1, the chiral pair is AAB-ABB. For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA. For T(5,2)=6, the chiral pairs are AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB.
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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[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 12}, {k, 1, n}] // Flatten
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PROG
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(PARI) \\ here 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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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