Order Matters and Matters of Order:  Euler' s Triangle

The different sets in which things can be arranged are called permutations.  The number of permutations of n things taken n at a time equals n! total possibilities, or, the sum of the values in one of Euler Triangle's rows.  See below for more.

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Row n of Euler's Triangle, if considered to be originating "seed numbers", will sum (ad infinitum) to make series which then sum reflexively to next series.  The values of a power of n will be the series at the (n+1)th summation from the seeds of row n.  Euler row n exclusively sums to only a single nth power series and no other power's series.  (Euler's rows might be considered a "naming" of power series:  Row n names the power of n.  It might be said that first summations from row n show terms of "the power of n series" at lower, shell (or nexus number) levels and continued summations show terms at the xⁿ series and higher levels of summation.)

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The Pascal Triangle-based formula for the values of Euler's Triangle:

The kth term of the nth row [where, for the Euler triangle, the top "1" is row 1 and the first number on the left is the 1st term] equals:

The summation of j = 0 through (k-1) of: 

-1^j * Pascal Triangle's jth of row (n+1) [where, for the Pascal Triangle, row 1 is "1,1" and the "1st" of the row is the second number from the left] * (k-j)^n

Use of the formula looks like this:  for the kth of row n, the first k values (including the first "1") of Pascal Triangle's row (n+1) are terms that each multiply, in order, {k, k-1, k-2 . . . 1}.  After that, each of the terms, in order, multiply  {k^n, (k-1)^n, (k-2)^n . . . 1^n}.  Lastly, subtract the second term from the first, add the third, subtract the fourth, etc. in alternating, positive/negative valuations of the terms.  That equals the kth of row n in Euler's Triangle.

(Interestingly, if k = (n+1), the formula works out to equal 0 in a long way, which is correct because there are only n terms per nth Euler row.  There logically cannot be n+1 ascents in n values.)

See Math World.  (Be careful of confusion because of variation about what row n and the kth value are called.)

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Values of Euler's Triangle relate to its own previous values and can be fully defined recursively, given the first "1":

Call the kth term of Euler Triangle's row n:  A(n,k)

A(n,k)   =   (n - k + 1) * A(n-1, k-1) + k * A(n-1, k)

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The kth of row n gives the number of permutations of [1,2,3...n] having k permutation ascents also called permutation runs.  (See Math World)  Which means that the kth of row n says how many sets of series made from n terms it would be possible to make that have ascending values internally for k number of ascents.  For example, there are three "ascents" in this series:  3, 4, 8; 2, 5; 1, 6, 7.  This is only one of example of three ascents within eight numbers of which there are 4,292 other possible arrangements as defined in the 3rd of row 8 in Euler's Triangle.

Table of ascents occurring within permutations of sets of n.   a < b < c < d . . .

Sloane's Encyclopedia of Integer Series' number:  A008292

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