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A212958
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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}.
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0, 0, 1, 1, 0, 1, 12, 22, 0, 21, 22, 123, 131, 343, 0, 342, 343, 1234, 2531, 4664, 0, 1421, 3242, 4663, 12345, 58985, 0, 58984, 58985, 23456, 497531, 713306, 0, 137421, 276842, 436463, 575884, 713305, 713306, 1234567, 1810675, 2907844, 4002993, 6197531, 8367727
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OFFSET
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1,7
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COMMENTS
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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).
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).
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.
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.
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).
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LINKS
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EXAMPLE
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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.
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.
The exhaustive computer program featured in the link finds that in decimal notation those three pairs with the ratio 19/9 = 2.111... are:
(1) {114,54}; (2) {57,27}; (3) {228,108}
In base-4 notation, those 3 pairs of distinct permutations are:
(1) {1302, 0312}; (2) {0321,0123}; (3) {3210,1230};
Now we calculate the fourth row terms per the sequence's definition:
(1302-0123)/9 = 131; (0312-0123)/9 = 21; (0321-0123)/9 = 22; (0123-0123)/9 = 0; (3210-0123)/9 = 343; (1230-0123)/9 = 123;
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, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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