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Number of binary [ n,4 ] codes without 0 columns.
7

%I #30 Oct 06 2019 03:01:51

%S 0,0,0,1,4,11,27,63,134,276,544,1048,1956,3577,6395,11217,19307,32685,

%T 54413,89225,144144,229647,360975,560259,858967,1301757,1950955,

%U 2893102,4246868,6174084,8892966,12696295,17973092,25237467,35163431,48629902,66774760,91063984

%N Number of binary [ n,4 ] codes without 0 columns.

%C "We say that the sequence (a_n) is quasi-polynomial in n if there exist polynomials P_0, ..., P_{s-1} and an integer n_0 such that, for all n >= n_0, a_n = P_i(n) where i == n (mod s)." [This is from the abstract of Lisonek (2007), and he states that the condition "n >= n_0" makes his definition broader than the one in Stanley's book. From Section 5 of his paper, we conclude that (a(n): n >= 1) is a quasi-polynomial in n.] - _Petros Hadjicostas_, Oct 02 2019

%H Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_4.html">Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns</a>. [See column k=4.]

%H H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,4,2}.]

%H Petros Hadjicostas, <a href="/A034253/a034253.txt">Generating function for a(n)</a>.

%H Petr Lisonek, <a href="https://doi.org/10.1016/j.jcta.2006.06.013">Combinatorial families enumerated by quasi-polynomials</a>, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]

%H David Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H David Slepian, <a href="https://doi.org/10.1002/j.1538-7305.1960.tb03958.x">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.

%o (Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):

%o def A034253col(k, length):

%o G1 = PSL(k, GF(2))

%o G2 = PSL(k-1, GF(2))

%o D1 = G1.cycle_index()

%o D2 = G2.cycle_index()

%o f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)

%o f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)

%o f = f1 - f2

%o return f.taylor(x, 0, length).list()

%o # For instance the Taylor expansion for column k = 4 (this sequence) gives

%o print(A034253col(4, 30)) #

%Y Cf. A034254, A034344, A034346, A034347, A034348, A034349, A253186.

%Y Column k=4 of A034253 and first differences of A034358.

%K nonn

%O 1,5

%A _N. J. A. Sloane_.

%E More terms by _Petros Hadjicostas_, Oct 02 2019