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A275442
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Triangle read by rows: T(n,k) is the number of compositions without 2's and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).
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1
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1, 1, 1, 2, 2, 2, 3, 4, 4, 8, 5, 16, 7, 26, 4, 9, 44, 12, 12, 70, 32, 16, 108, 76, 21, 166, 156, 8, 28, 248, 308, 32, 37, 368, 572, 104, 49, 540, 1020, 288, 65, 784, 1768, 696, 16, 86, 1132, 2976, 1568, 80, 114, 1622, 4908, 3304, 304, 151, 2312, 7944, 6624, 960, 200, 3280, 12652, 12768, 2640, 32
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OFFSET
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0,4
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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/4).
Sum of entries in row n is A005251(n+1).
T(n,0) = A000931(n+5) (= number of palindromic compositions of n without 2's).
Sum_{k >= 0} k*T(n,k) = A275443(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^2+z^4-2tz^4). 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 5 is [3,4] because the compositions of 5 without 2's are 5, 113, 131, 311, 14, 41, and 11111, having asymmetry degrees 0, 1, 0, 1, 1, 1, and 0, respectively.
Triangle starts:
1;
1;
1;
2;
2,2;
3,4;
4,8;
5,16.
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MAPLE
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G := (1-z^2)/(1-z-z^2+z^4-2*t*z^4): 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 == 2]], 1]]], {n, 0, 20}] // 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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