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Revision History for A039754 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
Irregular triangle read by rows: T(n,k) = number of binary codes of length n with k words (n >= 0, 0 <= k <= 2^n); also number of 0/1-polytopes with vertices from the unit n-cube; also number of inequivalent Boolean functions of n variables with exactly k nonzero values under action of Jevons group.
(history; published version)
#54 by Alois P. Heinz at Sun Jan 29 17:51:09 EST 2023
STATUS

proposed

approved

#53 by Sean A. Irvine at Sun Jan 29 15:53:59 EST 2023
STATUS

editing

proposed

#52 by Sean A. Irvine at Sun Jan 29 15:52:59 EST 2023
MATHEMATICA

AC[d_Integer] := Module[{C, M, p}, (* from W. Y. C. Chen algorithm *) M[p_List] := Plus @@ p!/(Times @@ p * Times @@ (Length /@ Split[p]!)); C[p_List, q_List] := Module[{r, m, k, x}, r = If[0 == Length[q], 1, 2*2^IntegerExponent[LCM @@ q, 2]]; m = LCM @@ Join[p/GCD[r, p], q/GCD[r, q]]; CoefficientList[Expand[Product[(1 + x^(k *r))^((Plus @@ Map[MoebiusMu[k/#]*2^Plus @@ GCD[#*r, Join[p, q]]&, Divisors[k]])/(k*r)), {k, 1, m}]], x]]; Sum[Binomial[d, p]*Plus @@ Plus @@ Outer[M[#1] M[#2] C[#1, #2]*2^(d - Length[#1] - Length[#2]) &, P[p], P[d - p], 1], {p, 0, d}]/(d! 2^d)]; AC[0] = {1, 1};

STATUS

proposed

editing

Discussion
Sun Jan 29
15:53
Sean A. Irvine: Initially this changed confused me, but I've come to the conclusion that it is indeed correct, the bounds in the Name make it clear there really should be two values in the 0-th row.
#51 by Michel Marcus at Tue Jan 10 12:03:37 EST 2023
STATUS

editing

proposed

Discussion
Tue Jan 10
12:12
Tilman Piesk: I don't get it. Should we not all see the same? Exactly how many characters per line are allowed?
13:11
Andrew Howroyd: It might depend on many things such as browser and  size of screen. I'm using Chrome and have an enormous monitor and the published sequence https://oeis.org/A039754 wraps for me. It may be by just 1 char - given that wrapping behavior is unpredictable I have removed more than the absolute minimum, but it will still wrap on phones.
13:22
Andrew Howroyd: Also bare in mind the developer(s) of this site periodically change things such as the css, so even if we could establish an exact number of safe chars today it might not be true tomorrow. There is just no hard science to this - keen observers will also notice that there are periodic changes to the way the editor handles spaces and editing in general - it's all slightly fuzzy and probably always will be.
14:13
Tilman Piesk: This kind of thing is generally true about websites, but the OEIS is probably the lone exception. This site is clearly intended to look the same everywhere. The layout is probably inherited from ancient times, when this was done on typewriter. (Otherwise we would have HTML and Unicode, like everyone else.) I suppose the intended limit is 80 characters per line. It should be easy to fix the CSS. Yes, on a phone held vertically the lines will still wrap (unless we change white-space from pre-wrap to pre). But browsers should make no difference. BTW, this question on Stackoverflow seems related: https://stackoverflow.com/questions/55149140/why-is-a-monospace-font-set-to-80em-taking-up-less-than-80-columns-of-text
15:01
Andrew Howroyd: *** It should be easy to fix the CSS *** No. The layout semantics of css are so ambiguously and poorly defined that there is no hope anyone including the developers of the specification will ever understand the specification. Most developers just muddle through life with the aid of random experimentation, stack-overflow searches and testing against a selection of browsers to see what the css actually does. Nothing about css is easy to fix.
#50 by Michel Marcus at Tue Jan 10 12:03:29 EST 2023
LINKS

D. Condon, S. Coskey, L. Serafin, and C. Stockdale, <a href="http://arxiv.org/abs/1412.4683">On generalizations of separating and splitting families</a>, arXiv preprint arXiv:1412.4683 [math.CO], 2014-2015.

H. Harald Fripertinger, <a href="http://dx.doi.org/10.1023/A:1008248618779">Enumeration, construction and random generation of block codes</a>, Designs, Codes, Crypt., 14 (1998), 213-219.

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

STATUS

proposed

editing

#49 by Andrew Howroyd at Tue Jan 10 11:57:58 EST 2023
STATUS

editing

proposed

#48 by Andrew Howroyd at Tue Jan 10 11:56:41 EST 2023
EXTENSIONS

Add T(0, 1) = 1 inserted. (There are two 0-ary functions.) _- _Tilman Piesk_, Jan 10 2023

Discussion
Tue Jan 10
11:57
Andrew Howroyd: I've tightened up the table slightly to avoid each row wrapping to 2 lines.
#47 by Andrew Howroyd at Tue Jan 10 11:54:40 EST 2023
EXAMPLE

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 sums

0 1 1 2

1 1 1 1 3

2 1 1 2 1 1 6

3 1 1 3 3 6 3 3 1 1 22

4 1 1 4 6 19 27 50 56 74 56 50 27 19 6 4 1 1 402

STATUS

proposed

editing

#46 by Tilman Piesk at Tue Jan 10 08:52:15 EST 2023
STATUS

editing

proposed

#45 by Tilman Piesk at Tue Jan 10 08:42:37 EST 2023
DATA

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1, 1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1

OFFSET

0,78

EXAMPLE

0 1 1 2

EXTENSIONS

Add T(0, 1) = 1. (There are two 0-ary functions.) Tilman Piesk, Jan 10 2023

STATUS

approved

editing

Discussion
Tue Jan 10
08:50
Tilman Piesk: Row 0 contained only T(0, 0) = 1, representing the contradiction. But the tautology is also a 0-ary function. Therefore the entry T(0, 1) = 1 is also needed. This way the row sums make sense again. A000616 is 2, 3, 6, 22... (not 1, 3, 6, 22...). When Neil started this sequence in 1999 it has 5 ones at the beginning, i.e. it started with row 1 instead of row 0. Later row 0 was added incompletely.