Talk:Factoradic

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Have received information from Peter Cameron to the effect that he first heard about factoradics from a talk attended in the mid-1980's, when the speaker used the term as if it was already standard. So without a source from that era, the best one can say is the origin is simply obscure. --J. W. McLeod 20:57, 29 Mar 2005 (UTC)

I think the problem may be that this is obvious, and has probably been invented several times. I once described it in Usenet as "factorial base" in January 2002[1], and I did not think it was anything special. Nor can I remember being told about it by anyone else before I used it myself to solve a minor problem. --Henry Bottomley 13:13, 3 Jun 2005 (UTC)

Donald E Knuth's The Art of Computer Programming provides additional information. In Volume 4, Fascicle 2: Generating All Tuples and Permutations the answer to question 7.2.1.2-3 on page 101 cites H. A. Rothe, Sammlung combinatorisch-analytischer Abhandlungen, 2 (1800), 263-264. He refers to inversion tables in question 5.1.1-7, and notes that he described the factorial number system in equation 4.1-(10). It also shows up earlier as the Algorithm P in section 3.3.2 Empirical Tests (for random numbers), pp 65-66 of Volume 2. Knuth also refers to a "Narayana cited in Section 7.2.1.7", which would be the history of combinatorics that is currently being drafted. So factoradics would seem to have been kicking around for over two centuries in one form or the other. --J. W. McLeod 15:05, 4 Jun 2005 (UTC)

Iverson was using it, presumably with the associated database applications, as a running example by the early 1970's. You may find Cantor Expansion to be a more productive term for investigating the history than factoradic. 83.76.113.61 23:57, 13 January 2007 (UTC)

[edit] Rightmost zero digit

At first look the last always zero digit in the definition seems redundant, but it's used in the algorithm for making a permutation from factoradic, so either the algorithm should be corrected or the last always zero digit should be present in the definition. Besides it fits nicely because you can just start counting from zero: the factoradic d_n...d₂d₁d₀ is equal to 0!d₀ + 1!d₁ + 2!d₂ + ... + n!d_n, where every digit d_i is in [0, i].

There is a major contradiction on this page, and I don't know how to solve it, so I'm hoping someone else will. At the beginning, and at the first table of numbers, a notation is used such that the nth radix is n!, but after that a notation is used such that the nth radix is (n-1)!. At the beginning 719 would be 54321, but after that it would be 543210. This is EXTREMELY confusing! Someone who actually knows about factoradic, please change this! Soon! —The preceding unsigned comment was added by 70.104.133.164 (talk • contribs).

I suggest that the rightmost always-zero digit should be suppressed from the definition. I give the following motivations:

1. Its useless.
2. It's confusing.
3. No serious reference is given supporting this convention; the OEIS reference which IS serious supports the other notation (without -0).
4. No other numbering system includes a digit which has the same value for all numbers.
5. In any positional numbering system using arabic digits, the number 1 is represented by 1.
6. In any positional numbering system using arabic digits, the sucessor of a number having least significant digit 0 is the number obtained by replacing that digit by 1.
7. No positional numbering system has (nor should have) two distinct places with the same weight, here d0=d1.
8. The only *vague* motivation for including the -0 is the "coding of permutations" issue, but for this one could easily adopt the definitions/algorithm to always add the -0 "explicitely".
9. The "explanation" that d0=0! since d0=b^0 for the usual positional system, is not valid. In fact, in the usual numbering system the weight of the least digit is 1 (which by accident happens to be equal to b^0), and then the weight of the other positions is obtained by sucessively multiplying the preceding weight by a number bigger than one.

I'm waiting for other people's opinion concerning this convention. Please don't hesitate to add a line below with remove (= remove trailing 0) or don't change (leave as is with trailing 0) or any other comment. — MFH:Talk 21:24, 27 March 2007 (UTC)

[edit] Non-integers

I was at the 0.999... page and it gave an example that "In the factoradic system, 1 = 1.000… = 0.1234…" . but the current Factoradic article doesn't seem to allow for a 'decimal' point. I would think that either the 0.999... article should have the factoradic example removed or the Factoradic article should be added to. It seems to me like there are different ways to implement a 'decimal' point though, and that having one could make there be multiple representations for the same number (1₁0₀.0₀1₁ would be another way of representing 2, right?) --Sgt. Muffles 18:56, 27 November 2006 (UTC)

I vaguely remember reading a magazine article about a factorial fraction number system. Integers were represented normally, but numbers after the fraction point a.(b)(c)(d)(e)(f)(g)(h)... were interpreted as

This factorial fraction system can represent any rational number exactly in a finite number of places after the fraction point (unlike the decimal system, which requires endlessly repeating patterns for things like 1/7).

For example,

 1/2 = 0.(1)
 1/3 = 0.(0)(2)
 1/4 = 0.(0)(0)(6)
 1/5 = 0.(0)(0)(0)(24)
 1/6 = 0.(0)(1)
 1/7 = 0.(0)(0)(0)(0)(0)(720)
 e   = 2.(1)(1)(1)(1)(1)(1)(1)(1)....
 1/e = 0.(1)(-1)(1)(-1)(1)(-1)(1)....

However, in this system, 0.(1)(2)(3)(4)... is significantly less than one. So perhaps the 0.999... page is referring to some *other* factoradic system? Or is it a typo? -- DavidCary --68.0.120.35 05:41, 17 December 2006 (UTC)

Let's take your definition for the time being (Sgt. Muffles seems to prefer an extra 0 to the right of the point but so long as the definition is clear it hardly matters much). I suspect there is a rule that fractional numbers should have the nth digit less than or equal to n and your examples should be:

1/2 = 0.1 or 0.023456789...

1/3 = 0.02 or 0.013456789...

1/4 = 0.012 or 0.011456789...

1/5 = 0.0104 or 0.010356789...

1/6 = 0.01 or 0.003456789...

1/7 = 0.003206 or 0.003205789...

e = 2.111111111...

1/e = 0.020406080...

and is indeed close to 1 --Henrygb 18:15, 17 December 2006 (UTC)

Ah, clever. How do we know that the nth digit will never need to be greater than n? (Is there something for factoradic, that would be analogous to Zeckendorf's theorem for Fibonacci coding ?) After thinking about this for a bit, I think I could make up a proof ... but that would be original research (WP:OR). -- DavidCary --68.0.120.35 22:14, 17 January 2007 (UTC)

It is a simple inductive proof that using --Henrygb 21:00, 18 January 2007 (UTC)