A275433 -id:A275433 - cp's OEIS frontend

A275442 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)).

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

Offset: 0

Emeric Deutsch, Aug 16 2016

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).

			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.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

John Tyler Rascoe, Rows n = 0..200, flattened
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A000931, A005251, A180177, A275433, A275443, A275446.

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

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 *)

T_zt(max_row) = {my(N = max_row+1, z='z+O('z^N), h=(1-z^2)/(1-z-z^2+z^4-2*t*z^4)); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_zt(10) \\ John Tyler Rascoe, May 09 2025

G.f.: G(t,z) = (1-z^2)/(1-z-z^2+z^4-2*t*z^4). In the more general situation of compositions into a[1]=1} z^(a[j]), 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.

A275434 Sum of the degrees of asymmetry of all compositions of n.

Original entry on oeis.org

0, 0, 0, 2, 4, 12, 28, 68, 156, 356, 796, 1764, 3868, 8420, 18204, 39140, 83740, 178404, 378652, 800996, 1689372, 3553508, 7456540, 15612132, 32622364, 68040932, 141674268, 294533348, 611436316, 1267611876, 2624702236, 5428361444, 11214636828

Offset: 0

Emeric Deutsch, Jul 29 2016

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry 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 degree of asymmetry is 0.

			a(4) = 4 because the compositions 4, 13, 22, 31, 112, 121, 211, 1111 have degrees of asymmetry 0, 1, 0, 1, 1, 0, 1, 0, respectively.

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A059570, A275433.

g := 2*z^3*(1-z)/((1-2*z)*(1-z-2*z^2)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
a := proc(n) if n = 0 then 0 elif n = 1 then 0 else -(4/9)*(-1)^n+(1/36)*(3*n-2)*2^n end if end proc: seq(a(n), n = 0 .. 32);

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[b[n - j, If[i==0, j, 0]] If[i > 0 && i != j, x, 1], {j, 1, n}]]];
a[n_] := Function[p, Sum[i Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0]];
a /@ Range[0, 32] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz in A275433 *)

G.f.: g(z) = 2*z^3*(1-z)/((1-2*z)*(1-z-2*z^2)). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).
a(n) = -(4/9)*(-1)^n + (3*n - 2)*2^n/36 for n>=2; a(0) = a(1) = 0.
a(n) = Sum_{k>=0} k*A275433(n,k).
a(n) = 2*A059570(n-2) for n>=3. - Alois P. Heinz, Jul 29 2016

cp's OEIS Frontend

A275442 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)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula