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).

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

Offset: 1

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• sign
• tabf

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[1],...,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).]

Examples

			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[1],...,a[4])= -1*a[4] + 2*a[1]*a[3] + 1*a[2]^2 - 3*a[1]^2*a[2] + a[1]^4.
Consider variables x_1, x_2, x_3, x_4, and let a[1] = Sum_i x_i, a[2] = Sum_{i,j} x_i*x_j, a[3] = Sum_{i,j,k} x_i*x_j*x_k, and a[4] = x1*x2*x3*x4, where in all the sums no term is repeated twice.
Then h(4; a[1],...,a[4]) = 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.
		

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.

Crossrefs

Cf. A000041, A000079, A036038, A036039, A036040, A036043, A048996, A103921, A105805, A115131, A210258.

Formula

The complete symmetric row polynomials h(n; a[1], ..., a[n]):= sum k over partitions of n of T(n, k)* A[k], with A[k] := a[1]^e(k, 1) * a[2]^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[1]^e(k, 1) * a[2]^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.

Extensions

Various sections edited by Petros Hadjicostas, Dec 15 2019