|
|
A276060
|
|
Triangle read by rows: T(n,k) is the number of compositions of n into parts congruent to 1 mod 3 and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/5)).
|
|
1
|
|
|
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 6, 3, 10, 5, 10, 4, 4, 20, 4, 7, 22, 12, 6, 34, 20, 10, 42, 36, 9, 64, 48, 8, 15, 70, 96, 8, 13, 112, 120, 32, 22, 124, 204, 56, 19, 184, 280, 112, 32, 212, 436, 176, 16, 28, 310, 564, 360, 16, 47, 346, 896, 504, 80, 41, 512, 1128, 920, 144, 69, 570, 1704, 1360
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
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/5).
Sum of entries in row n is A000930(n-1).
|
|
REFERENCES
|
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: G(t,z) = (1-z^2)(1+z+z^2)(1-z+z^2)(1+z -z^3)/(1-z^2-z^3+z^5-2tz^5-z^6+z^9). 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.
|
|
EXAMPLE
|
Row 7 is [2,4] because the compositions of 7 with parts in {1,4,7,10,... } are 7, 4111, 1411, 1141, 1114, and 1111111, having asymmetry degrees 0, 1, 1, 1, 1, and 0, respectively.
Triangle starts:
1;
1;
1;
1;
2;
1,2;
2,2.
|
|
MAPLE
|
G := (1-z^2)*(1+z+z^2)*(1-z+z^2)*(1+z-z^3)/(1-z^2-z^3+z^5-2*t*z^5-z^6+z^9): 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
|
|
MATHEMATICA
|
Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/5], 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_, ___} /; Mod[a, 3] != 1]], 1]]], {n, 0, 24}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|