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A276058
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Triangle read by rows: T(n,k) is the number of compositions of n with parts in {3,4,5,6,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/7)).
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1
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1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 6, 3, 10, 5, 14, 4, 24, 7, 30, 4, 6, 46, 8, 10, 58, 20, 9, 84, 36, 15, 106, 68, 13, 152, 112, 22, 188, 196, 19, 264, 304, 8, 32, 324, 492, 24, 28, 446, 732, 72, 47, 546, 1120, 160, 41, 744, 1616, 344
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OFFSET
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0,5
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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).
A sequence is palindromic if and only if its asymmetry degree i 0.
Number of entries in row n is 1 + floor(n/7).
Sum of entries in row n is A078012(n).
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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^2)(1-z +z^3)/(1-z-z^2+z^3-z^6+z^7-2tz^7). 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 7 is [1,2] because the compositions of 7 with parts in {3,4,5,...} are 7, 34, and 43, having asymmetry degrees 0, 1, and 1, respectively.
Triangle starts:
1;
0;
0;
1;
1;
1;
2;
1,2,
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MAPLE
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G := (1-z^2)*(1-z+z^3)/(1-z-z^2+z^3-z^6+z^7-2*t*z^7): 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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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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