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A215940
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Difference between the n-th and the first (identity) permutation of (0,...,m-1), interpreted as a decimal number, divided by 9 (for any m for which 10! >= m! >= n).
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7
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0, 1, 10, 12, 21, 22, 100, 101, 120, 123, 131, 133, 210, 212, 220, 223, 242, 243, 321, 322, 331, 333, 342, 343, 1000, 1001, 1010, 1012, 1021, 1022, 1200, 1201, 1230, 1234, 1241, 1244, 1310, 1312, 1330, 1334, 1352, 1354, 1421, 1422, 1441, 1444, 1452, 1454, 2100
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OFFSET
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1,3
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COMMENTS
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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.
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]
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)
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)
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LINKS
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FORMULA
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EXAMPLE
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The permutations of {0,1,2}, read as numbers, are {12, 21, 102, 120, 201, 210}. Subtracting the first one (12) from each of these numbers yields the differences {0, 9, 90, 108, 189, 198}. These are all multiples of 9, see comment and links. Dividing the differences by 9 yields {0, 1, 10, 12, 21, 22}, which are by definition the first six terms of this sequence.
Using all permutations of 0123 would yield 4!=24 terms, where the first 6 would be identical to those above. For n>10! we must consider permutations of (0,...,m-1) with m>10. These are no longer valid digits in base 10, and the numbers P(n) as defined in the comment are no longer equal to the concatenation. However, the first 10! terms obtained as (P(n)-P(1))/9, are still the same as for m=10;
In order to illustrate that the result is independent of the base chosen to make the calculation, let us consider permutations of 012 in base 3. The 3rd resp. 5th term ((102-012)/9=10 resp. (201-012)/9=21) would be ((1-0)*3^2+(0-1)*3+(2-2)*1)/(3-1) = 3 = 10[3], resp. ((2-0)*3^2+(0-1)*3+(1-2)*1)/(3-1) = 7 = 21[3]. The same terms would also be "10" and "21" if the calculation were made in base b=11. In that base, with digit "A" having the value b-1, we have (1023456789A - 0123456789A)/A = 0A000000000[11], (2013456789A - 0123456789A)/A = 02100000000[11], and (0A123456789 - 0123456789A)/A = 0A987654321[11] (the analog of (40123[5]-01234[5])/4[5] = 04321[5]). (End)
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MAPLE
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N:= 100: # to get a(1)..a(N)
for M from 3 while M! <= N do od:
p0:= [$1..M]: p:= p0: A[1]:= 0:
for n from 2 to N do
p:= combinat:-nextperm(p);
d:= p - p0;
A[n]:= add(10^(i-1)*d[-i], i=1..M)/9;
od:
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MATHEMATICA
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maxm = 5; Table[dd = FromDigits /@ Permutations[Range[m]]; (Drop[dd, If[m == 1, 0, (m - 1)!]] - First[dd])/9, {m, 1, maxm}] // Flatten (* Jean-François Alcover, Apr 25 2013 *)
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PROG
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(C) See links.
(PARI) A215940(n)=for(k=2, n+1, k!<n & next; k=vecsort( vector( (#k=vector(k, j, 10^j)~\10)!, i, numtoperm(#k, i)*k )); return((k[n]-k[1])/9)) \\ (It is of course more efficient to calculate a whole vector of the first k! terms.) - M. F. Hasler, Jan 12 2013
(PARI) first_n_factorial_terms(n)={my(u=n!); my(x=numtoperm(n, 0), y, z=vector(u), i:small); i=0; for(j=0, u-1, y=numtoperm(n, j)-x; z[i++]=fromdigits(vector(#x-1, k, vecsum(y[1..k])))); z} \\ R. J. Cano, Apr 19 2017
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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