cp's OEIS Frontend

The first 9! = 362880 terms of this sequence are permutations of the digits 1-9 with a(9!) = 987654321. - Jeremy Gardiner, May 28 2010

First differences are given in A209280 (for the first 9! terms) or in A219664 (for at least as much initial terms). - M. F. Hasler, Mar 03 2013

A230959(a(n)) = 0. - Reinhard Zumkeller, Nov 02 2013

After the first 9! terms, 8! + 7! = 9*7! of the initial terms are repeated with a leading '1' prefixed, cf. formula. However, a(9!+8!+7!) = 1219...3 is followed by 122...9 and permutations of the last 7 digits, before 12314..9. - M. F. Hasler, Jan 08 2020, corrected Aug 11 2022 thanks to a remark from Michael S. Branicky

Terms do not depend on the choice of m, provided that m!>n (the index of the considered term), and the numbers associated to a permutation s of {0,...,m-1} are N(s) = Sum_{i=1..m} s(i)*10^(m-i). This defines the present sequence for any arbitrarily large index, not limited to n <= 10!, for example.

Similar sequences might be built in another base b, they would always start (1, b-1, 2, b-1, 1, ...). The partial sums of this kind of sequence would yield the analog of A215940 in the corresponding base.

There are at least two palindromic patterns which are repeated throughout this sequence: one of them is "1,b-1,2,b-1,1" (It is optional here whether or not to include the 1's), another is built from the first 4!-1 terms (See the corresponding link for details).

Also, for 1<=n<=(9!)-1: The repeating parts in the first differences of A030299 divided by nine, i.e. a(n) = A219664(n)/9. - Antti Karttunen, Dec 18 2012. Edited by: R. J. Cano, May 09 2017

There are more palindromic patterns than those mentioned above: Similar to the first 3!-1 and the first 4!-1 terms, the first k!-1 terms are repeated for all other k>4. Frequent are also multiples of these, e.g., k*[1,9,2,9,1] = [2,18,4,18,2], [3,27,6,27,3], ...), [1, 19, 3, 8, 2, 67, 1, 29, 4, 7, 3, 176, 3, 7, 4, 29, 1, 67, 2, 8, 3, 19, 1], and others. The "middle part" of roughly half the length (e.g., [9,2,9] or [67,...,67] in the last example), is repeated even more frequently. - M. F. Hasler, Jan 14 2013

From R. J. Cano, Apr 04 2016: (Start)

Conjecture 1: Given 1A217626 and so on).

Lemma: Let P be an arbitrary set consisting of m integers; let x[i] be an element in P (with 1<=i<=m); let y[j] = x[j+1] - x[j] (with 1 <= j <= m-1) be the 1st differences of P. These differences are symmetric if y[j]=y[m-j] which for P implies the condition x[j]+x[m-j+1]=x[j+1]+x[m-j];

Consequence: When m=n! and P is a set with all the permutations for the letters 0..n-1, the preceding lemma implies P has associated at least a set Q such that 1st differences in Q are symmetric.

Generating algorithm: Such Q can be built based upon P and the condition given by the preceding lemma if it is removed from P (until P becomes empty) its 1st element tau, inserting them both in Q tau and its arithmetic complement to repdigit (n-1)*111...1 (n times 1) removing the mentioned complement from P.

Conjecture 2: The autosimilarity shown by a(n) is a consequence of the fact that the corresponding P is the set of the n! permutations in increasing sequence for the letters 0..n-1, and Q=P (it holds if they are replaced "a(n)" and "increasing" respectively with "-1*a(n)" and "decreasing").

Note: "Q=P" is a necessary but not sufficient condition for observing the autosimilarity in a(n).

Application: The "generating algorithm" described previously might be potentially useful for parallel computing. In combination with the partition scheme proposed at links in A237265, and multiple indirection. For example notice that in such sense an algorithm for generating k! permutations with an increasing sequence would require only k!/2 iterations because the other half would be already determined by symmetry.

Conjecture 3: For n>2, given P the set of permutations in increasing sequence for the letters 0..n-1, there are distributed with a symmetric pattern among its (n!)! permutations all those A000165(n!\2) of them such that their 1st differences are symmetric. Moreover by setting to zero the other elements whose 1st differences are not symmetric, we obtain an antisymmetric sequence.

(End)

Conjecture 4: If 2<=mR. J. Cano, Apr 19 2017

Consider the first y!-1 terms for even y; The central term a(y!/2) is determined by the difference between the (y/2+1)th row from the y-th matrix defining the irregular table in A237265 and the consecutive permutation preceding it in lexicographic order (See EXAMPLE). - R. J. Cano, May 09 2017

First 5!-1 terms are identical to A107346, and the 9!-1 terms are identical to A209280. (Updated by M. F. Hasler, Jan 12 2013, Mar 03 2013)

Because of the self-similar nature of A220664, this sequence can be also constructed by picking appropriate terms from there with the auxiliary sequence A220655, cf. formula.

Similarly, differences between successive permutations of {1,2,...,k} in lexicographic order interpreted as decimal numbers, for any k=2..9, produce the first (k!)-1 terms of this sequence. But for k=10 the result is ill-defined, so we can consider the sequence finite, well-defined only for n=1..362879. [See however the following comment. - Editor's note]

In sequence A030299 it is clearly defined how it extends beyond index n = 1!+2!+...+9! = A007489(9), so the sequence A220664 of its first differences is well-defined up to infinity. (The "result" mentioned above is ill defined because the meaning of "interpreted" is not clear.) But the preceding comment is misleading by speaking of "self similar nature", and the sequence definition as "repeating part" is also misleading: If the sequence is defined to be of finite length 9!-1 (thus equal to A209280), then it is indeed infinitely often repeated as a subsequence (of consecutive terms) in A220664 (even when the latter was defined using concatenation for permutations of more than 9 elements, but then not as differences of the terms following 12345678910 where it was expected, but, e.g., as differences of the terms following 10123456789, etc.).

Since A030299 has been defined through a ("simpler") sum rather than concatenation, the nice mathematical properties of A220664, and even more this sequence A219664, persist beyond n=9!. - M. F. Hasler, Jan 12 2013

Original definition: "Quotients of the polynomial remainder theorem for Diophantine equations among permutations."

The set built from the first terms of this sequence {0, 1, 10, 12, 21, 22, ....} contains the general solutions for a class of Diophantine linear equations among permutations, if one takes into account the unlimited number of distinct bases where these may be read.

From M. F. Hasler, Jan 12 2013, edited by R. J. Cano, May 08 2017: (Start)

Let P be the sequence of permutations of (0,...,m-1) interpreted as decimal numbers, P(n) = Sum_{i=1..m} 10^(m-i)*s(i) where s=(s(1),...,s(m)) is the n-th permutation (in lexicographical order), n <= m!. Then the difference P(n)-P(1) is independent of the choice of m, and divisible by 9. (Since 10 == 1 (mod 9), the numbers P(n) are all congruent (mod 9) to the sum s(1)+...+s(m).) This yields well-defined terms a(n)=(P(n)-P(1))/9.

For n>10!, P(n) will no longer be the concatenation of the "digits" (some of which will exceed 9). The pattern present in the decimal representation of the first terms will also be lost since there will be digits as large as d+1.

Note that the same a(n) is obtained independently of the chosen base b, provided that (i) 10 and 9 in the above are replaced with b and b-1, (ii) the result is (and can be) written in base b. (This implies the restriction to terms which can be written using the digits 0-9 to which the OEIS is limited.) See EXAMPLES for an illustration. (End)

We have P(n)-P(1)=a(n)*g(n), with g(n) = 9 = 10-1. Considering this and P(n) as polynomials in x=10, one can see an analogy with the polynomial remainder theorem. [Given as "formula" by R. J. Cano, rephrased by M. F. Hasler, Jan 12 2013]

Contribution by R. J. Cano, Feb 09 2013, (Start)

The maximum of the first m! terms of this sequence is given in base R by the explicit formula (please see A211869): max(m,R)=Sum_{k=1..m} k*(m-k)*R^(m-k-1);

If the first m! terms of this sequence were computed reading the permutations in base A033638(m), dividing their differences by A033638(m)-1, the resulting quotients would be written in the same way (with the same digits) in every base > A033638(m).

(End)

From R. J. Cano, Apr 29 2016, (Start)

Although in the sequence name it reads: "permutation of (0,...,m-1)", the most general statement that could replace it is: "permutation of any m-tuple of integers all of them in arithmetic progression", obtaining a multiple of this sequence, lambda*a(n), where lambda is the common difference for the progression. It works in such way because only the differences is what matters here.

Given x>1 and k>=0, if a polynomial G(x) of degree k is divided by x-1 then the remainder will be the sum of all the coefficients in G. Let us consider the case in which those coefficients are the differences among the letters ("digits") of two permutations for the same set of letters (0..x-1): The sum of all those differences must vanish. This explains how the difference between two of such permutations expressed in base x is 0 mod x-1, particularly why differences for pairs of permutations are divisible by 9.

Another way of introducing this sequence takes advantage of the fact that for n>1, n! is even. Consider for n>1 to obtain only the first n! terms. This can be done by subtracting the last permutation from the first, the penultimate permutation from the second, and so on by following the pattern (P(k)-P(n!-k+1))/9 with 1<=k<=n!. Such procedure generates an antisymmetric sequence f(k) from which a(k)=(f(k)+f(n!))/2. This partially explains why A217626 is symmetric. Also a base-independent treatment is possible using linear algebra: Column vectors and the strictly lower triangular matrix instead of the division by (r-1) where r is the base (and r=10 here for this sequence). This approach leads one to conclude that terms in this sequence are the differences between pairs of vectors made from the first n-1 partial sums of letters ("digits") taken from permutations for n consecutive letters, when components in these vectors are viewed as coefficients for a power series in r=10.

(End)

We can produce similar sequences of length n!-1 from all the n-set permutations (1,...,n), starting from n=2 up to n=9. The next larger sequence contains always the preceding sequence as its proper prefix. See A219664 for the largest such sequence. - Antti Karttunen, Dec 18 2012

See A209280 for the extension of this sequence to 9!-1 terms, and for comments and formulas which apply to this subsequence. - M. F. Hasler, Jan 15 2013

From M. F. Hasler, Jan 12 2013: (Start)

Note [updated Mar 03 2013]: The definition of sequence A030299 has been slightly modified in Jan. 2013, and as a consequence the following properties remain valid beyond the first A007489(9)-1 = 409112 terms, which had not been the case before, when A030299 had been defined through concatenation of the lexicographically ordered permutations, which in case of elements >= 10 broke up the nice mathematical properties (esp. of the sequence A219664 = 9*A217626 cited below).

This sequence taken modulo 9 is zero except (possibly) at indices where a run of permutations ends in A030299. (These indices are given by A007489(n), n>0.) There it equals (mod 9) the "n" of the following run. E.g., a(1)=2 (mod 9), and A030299(1+1)=12 is the start of the run for n=2; a(3)=3 (mod 9) and A030299(3+1)=123 is the start of the run for n=3, a(9)=4 (mod 9) and A030299(9+1)=1234 is the start of the run for n=4, etc.

The subsequence between these indices (A007489(n)+1,...,A007489(n+1)-1), always starts with the same terms, listed in A219664 = 9*A217626 (= A209280 = A107346 where the latter are defined). (End)