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A275444
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Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,2,3} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).
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1
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1, 1, 2, 2, 2, 3, 4, 3, 10, 6, 14, 4, 6, 26, 12, 11, 34, 36, 11, 62, 68, 8, 20, 82, 140, 32, 20, 144, 228, 112, 37, 186, 424, 264, 16, 37, 316, 664, 608, 80, 68, 404, 1176, 1168, 320, 68, 676, 1784, 2312, 896, 32, 125, 860, 3032, 4096, 2304, 192
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OFFSET
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0,3
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COMMENTS
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The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
number of entries in row n is 1 + floor(n/3).
Sum of entries in row n is A000073(n+2).
T(2n,0) = T(2n+1,0) = A001590(n+3).
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REFERENCES
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S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
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LINKS
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FORMULA
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G.f.: G(t,z) = (1+z)(1+z^2)/(1-z^2 -z^4 -z^6 -2tz^3*(1+z+z^2 )). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, we have G(t,z) =(1 + F(z))/(1 - F(z^2) - t(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.
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EXAMPLE
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Row 4 is [3,4] because the compositions of 4 with parts in {1,2,3} are 13, 31, 22, 211, 121, 112, and 1111, having asymmetry degrees 1, 1, 0, 1, 0, 1, and 0, respectively.
Triangle starts:
1;
1;
2;
2,2;
3,10;
6,14,4.
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MAPLE
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G := (1+z)*(1+z^2)/(1-z^2-2*t*z^3-(1+2*t)*z^4-2*t*z^5-z^6): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
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MATHEMATICA
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Table[BinCounts[#, {0, 1 + Floor[n/4], 1}] &@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 3]], 1]]], {n, 0, 17}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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