|
%I #10 Aug 23 2016 04:46:34
%S 1,1,1,1,2,1,2,2,2,1,4,3,4,2,8,4,6,4,3,12,4,6,8,12,4,20,12,8,14,28,5,
%T 32,24,8,11,24,52,8,7,52,40,32,15,38,96,32,10,80,72,88,21,56,172,80,
%U 16,14,122,132,192,16,29,84,304,160,80,19,184,232,392,80,40,128,516,312,256,26,278,384,768,240,32
%N Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,4} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).
%C 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).
%C Number of entries in row n is 1 + floor(n/5).
%C Sum of entries in row n is A003269(n+1).
%C T(n,0) = A226517(n+10)
%C Sum(k*T(n,k), k>=0) = A276063(n).
%D S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.
%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%F G.f.: G(t,z) = (1+z +z^4)/(1-z^2-2tz^5-z^8). 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.
%e Row 6 is [2,2] because the compositions of 6 with parts in {1,4} are 411, 141, 114, and 111111, having asymmetry degrees 1, 0, 1, and 0, respectively.
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1;
%e 2;
%e 1, 2;
%e 2, 2;
%e 1, 4.
%p G := (1+z+z^4)/(1-z^2-2*t*z^5-z^8): 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
%t Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/4], 1}], # != 0 &] &@ 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_, ___} /; Nor[a == 1, a == 4]]], 1]]], {n, 0, 25}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)
%Y Cf. A003269, A226517, A276063.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Aug 22 2016
|
