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# User talk:Wolfdieter Lang

While I would like to say no problem with the quoted comment below, I am a bit confused. Maybe you meant something on the order of pi is "not" the prime index which is also the prime count? I think I read where that was the reason why pi(x) was used as to shorten prime index to to one greek letter to describe the prime count. I agree with the rest of your statements. I am always more worried about my writing leading to confusion than commuication so I understand the desire to get things strait.

"Wolfdieter Lang: Thanks John W. Nicholson. Sorry, I was reading pi for 'prime index', but it doesn't matter. Anyway, its good to say what pi means. WL"

John W. Nicholson 18:54, 16 May 2014 (UTC)

Dear Mr. Nicholson, the 'sorry' was because I did not realize in my first comment that it didn't matter whether pi is read as 'prime index' or as prime counting function pi(x), for x a prime. So my original comment should have been something like: 'Please give the definition of pi'. Best regards, Wolfdieter Lang

## A115131 : Waring numbers for power sums functions in terms of elementary symmetric functions.

dear Wolfdieter,

it was a pleasant surprise to find A115131 already present in the OEIS. The reference to "Waring numbers" sent me to google for them, but I only found references to Lagrange's 'four square' theorem. Question: what prompted you to use that name?

I stumbled on A115131 by substituting the n-th roots of 1 into the n variables of the monomial symmetric functions. in reverse lexicographic ordering:

{1}, {2,-1}, {3,-3,1}, {4,-4,-2,4,-1}, {5,-5,-5,5,5,-5,1}, {6,-6,-6,6,-3,12,-6,2,-9,6,-1}, {7,-7,-7,7,-7,14,-7,7,7,-21,7,-7,14,-7,1}, {8,-8,-8,8,-8,16,-8,-4,16,8,-24,8,8,-12,-24,32,-8,-2,16,-20,8,-1}, {9,-9,-9,9,-9,18,-9,-9,18,9,-27,9,9,18,-27,-27,36,-9,3,-27,18,-9,54,-45,9,9,-30,27,-9,1},{10,-10,-10,10,-10,20,-10,-10,20,10,-30,10,-5,20,20,-30,-30,40,-10,10,-15,10,-60,40,-10,60,-50,10,-10,-15,60,-25,40,-100,60,-10,2,-25,50,-35,10,-1}

All row sums equal 1. (Check: sum over monomials is just s_n : Schur of the one-part partition {n}).

Doing the analogous thing for the functions e, h, p, m, f and s shows (proof?) that: e(lambda,n)= -(-1)^n if lambda={n} else 0; h(lambda,n)= 1 if lambda={n} else 0; p(lambda,n)= n if lambda={n} else 0; s(lambda,n)= (-1)^(k+n) if lambda={k,1^(n-k)} else 0; f(lambda,n)= -(-1)^n m(lambda,n) and finaly m(lambda,n) is the tricky one, being -(-1)^n * A115131.

About the formula you gave: a(n, k)= (n/m(n, k))*|A111786(n, k)| for the k-th partition of n with m(n, k) parts in the Abramowitz-Stegun order. n>=1, k=1, ..., I recognise the matrix A111786 as the 'involution omega' e<->h (Mcdonald), but just dont't get it. The expression m(n,k) is obscure to me. Could you point me to it? If I get it right, then I'll enter a Mathematica %t line (for other users with that 'handicap') ;-)) Wouter Meeussen 15:28, 28 February 2015 (UTC)

From Wolfdieter Lang, Mar 09 2015 to Wouter Meeussen: The name Waring numbers was chosen because this is Waring's formula for N indeterminates. see. e.g.,http://planetmath.org/waringsformula. It is the solution of the Newton formulas, and expresses the sum over the n-th power of the indeterminates x1, ..., xN in terms of the elementary symmetric functions of these indetermiantes. See also the quoted references under A115131. In my paper I called this Girard's formula and Girard-Waring formula. For the history see my page http://www.itp.kit.edu/~wl/links.html (further down, the year 2006: Albert Girard and the Waring formula). This is a N-variable generalization of Chebyshev's T-polynomials (see the quoted Lidl and Lidl-Wells papers). The array is a partition array where the partitions are ordered like in Abramowitz-Stegun (A-St order, also known as Hindenburg order, see A036036, my comment from Apr 04 2011). The polynomial N*t^{(n)} (si1,...siN) (sij for \sigma_j, the j-th elementary symmetric functions of indeterminates x1, ..., xN (where the superscript N has been omitted at the si's)) has the coefficient a(n,k) = A115131(n, k) multiplying the sigmas corresponding to the k-th partition of n in the A-St order. See the given example for n=4, e.g., N*T^{(N)}_4(si1,...,siN) = .... -4*s1^2*si2 + ... , because in row n=4 [-4 4 2 -4 1] the 4-th partition of n=4 is in A-St order is (1^2,2), corresponding to si1^2*si2. The notation m(n, k) in the formula for a(n, k) means the number of parts of the k-th partition of n in A-St order. See the partition array A036043.