

A076832


Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).


8



1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 7, 19, 30, 35, 36, 1, 8, 29, 56, 73, 79, 80, 1, 10, 44, 107, 161, 186, 193, 194, 1, 12, 66, 200, 363, 462, 497, 505, 506, 1, 14, 96, 372, 837, 1222, 1392, 1439, 1448, 1449, 1, 16, 136, 680, 1963, 3435, 4282
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OFFSET

1,3


COMMENTS

It seems that Harald Fripertinger at his website defines T(n,k) = T(n,n) for k > n (and thus he gets an orthogonal array). It seems that T(n,n) = A034343(n).
It seems that T(n,k=2) = A001399(n) for n >= 2 (with A001399(n=1) = T(1,1)); T(n,k=3) = A034337(n) for n >= 3 (with A034337(n) = T(n,n) for 1 <= n <= 2); T(n,k=4) = A034338(n) for n >= 4 (with A034338(n) = T(n,n) for 1 <= n <= 3); and so on. See the Crossrefs below for more information.
To get the g.f. of column k (starting at n = 0 with T(n=0,k) := 1 rather than at n = k), modify the Sage program below (cf. function f).
(End)


LINKS

Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used to compute T_{nk2} using the cycle index of PGL_k(2).]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194204. [Apparently, the notation for T(n,k) is T_{nk2}.]


EXAMPLE

Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
1;
1, 2;
1, 3, 4;
1, 4, 7, 8;
1, 5, 11, 15, 16;
1, 7, 19, 30, 35, 36;
1, 8, 29, 56, 73, 79, 80;
1, 10, 44, 107, 161, 186, 193, 194; ...


MAPLE

# We illustrate how to get a g.f. for column k in Maple when k is small.
with(GroupTheory);
G := ProjectiveGeneralLinearGroup(4, 2);
GroupOrder(G);
# We get that the order is 20160.
f:=CycleIndexPolynomial(G, [x(1..20160)]);
# We get
# 1/20160*x1^15 + 1/192*x1^7*x2^4 + 1/96*x1^3*x2^6 + 1/16*x1^3*x2^2*x4^2 +
# 1/18*x1^3*x3^4 + 1/6*x1*x2*x3^2*x6 + 1/8*x1*x2*x4^3 + 1/180*x3^5 + 2/7*x1*x7^2 +
# 1/12*x3*x6^2 + 1/15*x5^3 + 2/15*x15
# The only dummy variables that appear are x1, x2, x3, x4, x5, x6, x7, and x15.
g:=subs(x1 = 1/(1  y), subs(x2 = 1/(y^2 + 1), subs(x3 = 1/(y^3 + 1), subs(x4 = 1/(y^4 + 1), subs(x5 = 1/(y^5 + 1), subs(x6 = 1/(y^6 + 1), subs(x7 = 1/(y^7 + 1), subs(x15 = 1/(y^15 + 1), f))))))));
# Then we take a Taylor expansion of the above g.f.
taylor(g, y=0, 50);
# We get a Taylor expansion for column k = 4 (i.e., A034338).


PROG

(Sage) # Fripertinger's method to find the g.f. of column k for small k:
def A076832col(k, length):
G = PSL(k, GF(2))
D = G.cycle_index()
f = sum(i[1]*prod(1/(1x^j) for j in i[0]) for i in D)
return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives A034338:


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



