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 A111786 Array used to obtain the complete symmetric function in n variables in terms of the elementary symmetric functions; irregular triangle T(n,k), read by rows, with n >= 1 and 1 <= k <= A000041(n). 7
 1, -1, 1, 1, -2, 1, -1, 2, 1, -3, 1, 1, -2, -2, 3, 3, -4, 1, -1, 2, 2, 1, -3, -6, -1, 4, 6, -5, 1, 1, -2, -2, -2, 3, 6, 3, 3, -4, -12, -4, 5, 10, -6, 1, -1, 2, 2, 2, 1, -3, -6, -6, -3, -3, 4, 12, 6, 12, 1, -5, -20, -10, 6, 15, -7, 1, 1, -2, -2, -2, -2, 3, 6, 6, 3, 3, 6, 1, -4, -12, -12, -12, -12, -4, 5, 20, 10, 30, 5, -6, -30, -20, 7, 21, -8, 1, -1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The unsigned numbers give A048996. They are not listed on pp. 831-832 of Abramowitz and Stegun (reference given in A103921). One could call these numbers M_0 (like M_1, M_2, M_3 given in A036038, A036039, A036040, resp.). The sequence of row lengths is A000041(n) (partition numbers). The sign is (-1)^(n + m(n,k)) with m(n,k) the number of parts of the k-th partition of n taken in the mentioned order. For m(n,k), see A036043. The row sum is 1 for n = 1, and 0 otherwise. The unsigned row sum is 2^(n-1) = A000079(n-1) for n >= 1. The complete symmetric polynomial is also h(n; a,...,a[n]) = Det(A_n) with the matrix elements of the n X n matrix A_n given by A_n(k, k+1) = 1 for 1 <= k < n, A(k, m) = a[k-m+1] for n >=  k >= m >= 1, and 0 otherwise. [For an explanation of this statement, see the example for n = 4 below. See also p. 3 in MacMahon (1960).] REFERENCES V. Krishnamurthy, Combinatorics, Ellis Horwood, Chichester, 1986, p. 55, eqs. (48) and (50). LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Wolfdieter Lang, First 10 rows. P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 4. OEIS, Orderings of partitions. FORMULA The complete symmetric row polynomials h(n; a, ..., a[n]):= sum k over partitions of n of T(n, k)* A[k], with A[k] := a^e(k, 1) * a^e(k, 2) * ... * a[n]^e(k, n) is the k-th partition of n, in Abramowitz-Stegun order (see A105805 for this reference), is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)], for k = 1..p(n), where p(n) = A000041(n) (partition numbers). G.f.: A(x) = 1/(1 + Sum_{j = 1..infinity} (-1)^j * a[j]). T(n, k) is the coefficient of x^n and a^e(k, 1) * a^e(k, 2) * ... * a[n]^e(k, n) in A(x) if the k-th partition of n, counted using the Abramowitz-Stegun order, is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] with e(k, j) >= 0 (and if e(k, j) = 0 then j^0 is not recorded). T(n, k) = (-1)^(n + m(n, k)) * m(n, k)!/(Product_{j = 1..n} e(k, j)!), where m(n, k) := Sum_{j = 1..n} e(k, j), with [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] being the k-th partition of n in the mentioned order. Here m(n, k) is the number of parts of the k-th partition of n. For m(n,k), see A036043. EXAMPLE Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:    1;   -1,  1;    1, -2,  1;   -1,  2,  1, -3,  1;    1, -2, -2,  3,  3, -4,  1;   -1,  2,  2,  1, -3, -6, -1, 4, 6, -5, 1,    ... h(4; a,...,a)= -1*a + 2*a*a + 1*a^2 - 3*a^2*a + a^4. Consider variables x_1, x_2, x_3, x_4, and let a = Sum_i x_i, a = Sum_{i,j} x_i*x_j, a = Sum_{i,j,k} x_i*x_j*x_k, and a = x1*x2*x3*x4, where in all the sums no term is repeated twice. Then h(4; a,...,a) = Sum_i x_i^4 + Sum_{i,j} x_i^3*x_j + Sum_{i,j} x_i^2*x_j^2 + Sum_{i,j,k} x_i^2*x_j*x_k + Sum_{i,j,k,m} x_i*x_j*x_k*x_m, where again in all the sums no term is repeated twice. Thus, indeed, h is the complete symmetric polynomial in four variables x_1, x_2, x_3, x_4. CROSSREFS Cf. A000041, A000079, A036038, A036039, A036040, A036043, A048996, A103921, A105805, A115131, A210258. Sequence in context: A210961 A250007 A048996 * A072811 A296559 A233548 Adjacent sequences:  A111783 A111784 A111785 * A111787 A111788 A111789 KEYWORD sign,tabf AUTHOR Wolfdieter Lang, Aug 23 2005 EXTENSIONS Various sections edited by Petros Hadjicostas, Dec 15 2019 STATUS approved

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Last modified April 19 21:57 EDT 2021. Contains 343117 sequences. (Running on oeis4.)