cp's OEIS Frontend

A212958 Array with a variable number of columns, where terms in the n-th row are the differences (computed in decimal base and divided by 9) between equal ratio permutations, found in the base n>=2, and the first (in ascending order of digits) minimal value permutation of {0,1,...,n}.

%I A212958 #64 Sep 10 2018 15:03:23
%S A212958 0,0,1,1,0,1,12,22,0,21,22,123,131,343,0,342,343,1234,2531,4664,0,
%T A212958 1421,3242,4663,12345,58985,0,58984,58985,23456,497531,713306,0,
%U A212958 137421,276842,436463,575884,713305,713306,1234567,1810675,2907844,4002993,6197531,8367727
%N A212958 Array with a variable number of columns, where terms in the n-th row are the differences (computed in decimal base and divided by 9) between equal ratio permutations, found in the base n>=2, and the first (in ascending order of digits) minimal value permutation of {0,1,...,n}.
%C A212958 It is conjectured (conjecture #1 - based on the observation of the programmatical exhaustive computation results) that among all possible distinct {0,1,...,n} permutations for each base n>=2, there are at least two pairs of such permutations that yield the same ratio, equal to A221740(p)/A221741(p) = (p^2*(p+1)^p -(p+1)^p+1)/(-p^2+p*(p+1)^p+(p+1)^p-p-1) where p=n-1. Additionally it is conjectured (conjecture #2) that the number of such pairs for each n is equal to A039649(p).
%C A212958 Construction of the terms for the n-th row of this sequence (where n is the base, n>=2) requires pre-calculation of the (mentioned in the above conjecture #1) permutation pairs (see the StackExchange link) that have the same ratio. Given the known (as defined above) ratio, actual values of the permutation pairs for each n (there are A039649(p) such pairs for each n>=2 - as conjectured above) are findable either by trial and error, or via executing an exhaustive computer program (see the link, featuring PARI/GP program - detection script for finding the equal-ratio permutation pairs for each base in the range from 2 to 10).
%C A212958 The terms of this sequence should be viewed in the table layout. The above mentioned table is an array with a variable number of columns. The number of terms (columns)in n-th row is, as conjectured above, equal to 2*A039649(p). Each n-th row (rows are counted from n=2) starts with a zero term because (as conjectured - conjecture #3) the 1st in ascending order minimal value permutation of {0,1,...,n} is always present among elements of the pairs, and, as it appears, always ends with the a(n=sum(i=2...n,2*A039649(i))) term, which has a nonzero value, and always seems to be equal to A215940((n)!) - conjecture #4.
%C A212958 Denoting the decimal value of the number made by the concatenation of the digits of the first (identical) base-n permutation of {0,1,...,n} as Pn, and considering terms of this sequence a(n) as members of a two-dimensional array b(n,k), where n is the row number and k is the position number in the n-th row (n>1, 1 <= k <=2*A039649(p)), then (per conjecture #1) it is true that in each array row "n" there are A039649(p) pairs of decimal integer values j and l such that (b(n,j)+ Pn)/(b(n,l)+ Pn) = A221740(p)/A221741(p) for n>=2, 1 <= j <= A039649(p), 1 <= l <= j.
%C A212958 It also appears that in the "next" (n+1)-th row there is always present a term that is less by one in decimal value as compared with the ending term in the "previous" n-th row (conjecture #5).
%H A212958 R. J. Cano, <a href="https://oeis.org/w/images/b/b8/PovolotskyPairs.gp.txt">PARI/GP program - detection script for finding the equal ratio permutation pairs for each base in the range from 2 to 10</a>
%H A212958 A. Povolotsky, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;7a24ab66.1301">Number of specific permutations pairs relates to Euler Phi totient function ?</a>, NMBRTHRYY 27 Jan 2013
%H A212958 StackExchange, <a href="http://math.stackexchange.com/questions/210578">Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0</a>.
%e A212958 For the fourth (n=4) row, which relates to base-4 four-digit {0, 1, 2, 3} distinct permutations, there are A039649(p) pairs where p = n-1 and thus for n=4, p=3, A039649(3)=3 - so there are three pairs in the fourth row.
%e A212958 Those pairs are supposed to have the same ratio, which can be calculated using the expression A221740(p)/A221741(p) = (p^2*(p+1)^p - (p+1)^p+1)/(-p^2 + p*(p+1)^p + (p+1)^p - p - 1), which for n=4 (p = n-1 = 3) yields 19/9 = 2.111.
%e A212958 The exhaustive computer program featured in the link finds that in decimal notation those three pairs with the ratio 19/9 = 2.111... are:
%e A212958   (1) {114,54}; (2) {57,27}; (3) {228,108}
%e A212958 In base-4 notation, those 3 pairs of distinct permutations are:
%e A212958   (1) {1302, 0312}; (2) {0321,0123}; (3) {3210,1230};
%e A212958 Now we calculate the fourth row terms per the sequence's definition:
%e A212958   (1302-0123)/9 = 131; (0312-0123)/9 = 21; (0321-0123)/9 = 22; (0123-0123)/9 = 0; (3210-0123)/9 = 343; (1230-0123)/9 = 123;
%e A212958 Thus, for the fourth row (n=4), which corresponds to base 4 (note that rows in the table are counted starting with n=2, which corresponds to base 2) we get the following 6 (three pairs) sequence terms, presented as sorted in ascending order: 0, 21, 22, 123, 131, 343, ...
%Y A212958 Cf. A215940, A221740, A221741.
%K A212958 nonn,tabl,base
%O A212958 1,7
%A A212958 _Alexander R. Povolotsky_, Feb 14 2013