A115131 -id:A115131 - cp's OEIS frontend

A308684 Partition array T(n, k) for the coefficients of the n-th power sums of the second elementary symmetric function in terms of the elementary symmetric functions.

1, 2, -2, 1, 3, -3, -3, 3, 3, -3, 1, 4, -4, -4, -4, 6, 4, 8, -8, -4, 4, -4, 4, 2, -4, 1, 5, -5, -5, -5, -5, 10, 5, 10, 10, -15, 5, -15, 5, 5, -5, -15, 10, 5, 10, -5, -5, -5, 5, 5, 5, -5, 5, 5, -5, 1

Offset: 1

Wolfdieter Lang, Jul 08 2019

The length of row n is A209816(n) (number of partitions of 2*n with at most n parts).
This is a generalization of the Girard-Waring array A115131.
In A324254 the general definition psigma(n, r) has been given for the r-th power sum of the n-th elementary symmetric function. There it is given in terms of the ordinary power sums {ps(j*r)}_{j=1..n}. Here psigma(2, n) = (1/2)*(-ps(2*n) + (ps(n))^2) is considered (see row n = 2 in A324254), and it is written in terms of elementary symmetric functions e_k(x1, x2, ...x_N), using the Girard-Waring formula for power sums ps. The number N >= 1 of indeterminates is suppressed in the notations.

			The irregular triangle (partition array) T(n, k)  begins:
n\k 1  2  3  4  5   6  7  8   9  10  11  12 13 14 15 ...
-------------------------------------------------------------------------------------------
1:  1
2:  2 -2  1
3:  3 -3 -3  3  3  -3  1
4:  4 -4 -4 -4  6   4  8 -8  -4   4  -4   4  2 -4  1
...
n = 5: [[5], [-5, -5, -5, -5, 10], [5, 10, 10, -15, 5, -15, 5, 5], [-5, -15, 10, 5, 10, -5, -5, -5, 5], [5, 5, -5, 5, 5, -5, 1]];
n = 6: [[6],[-6, -6, -6, -6, -6, 15], [6, 12, 12, 12, -24, 6, 12. -24, 6, -12, 12, 2], [-6, -18, -18, 18, 9, -18, 36, 0, 0, -18, 6, -18, 9, 0, 3], [6, 24, -12, -12, -18, 0, 0, 18, 12, 0, -12, -6, 6], [-6, 6, 6, 3,-6,-12, -2, 6, 9, -6, 1]];
n = 7: [[7], [-7, -7, -7, -7, -7, -7, 21] , [7, 14, 14, 14, 14, -35, 7, 14, 14, -35, 7, 7, -35, 14, 7, 7], [-7, -21, -21, -21, 28, 14, -21,-42, 56, 7, 28, 7, -21, -21, -7, 28, -21, 14, -21, 7, -7, -7, 14],  [7, 28, 28, -21, -21, 42, -63, -14, -7, -7, 35, 14, -21, 35, -14, 35, -14, -21, 7, -21, 14, -7, 7], [-7, -35, 14, 14, 7, 28, 7, 7, -21, -21,-21,-14, -7, 35, 7, 14, 7, -21, -7, 7], [7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1]]:
Brackets combine terms belonging to the same number of parts.
...
n = 3: psigma(2, 3) := Sum_{1<= i1 < i2 <= N} (x_{i1}*x_{i2})^3 = (1/2)*(-ps(2*3) + (ps(3))^2) = 3*e_6 - 3*e_1*e_5 - 3*e_2*e_4 + 3*(e_3)^2 + 3*(e_1)^2*e_4 - 3*e_1*e_2*e_3 + (e_2)^3. This becomes an identity if the e_j are written in terms of the indeterminates x_1, ..., x_N, for any N >= 1.

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].

Cf. A115121, A324254 (psigma(2, n) in terms of power sums).

psigma(2, n) = Sum_{k=1.. A209816(n)} T(n, k)*Product_{j=1..2*n} (e_j)^a(2*n,k,j), for n >= 1, if the k-th partition of 2*n (in Abramowitz-Stegun order) is Product_{j=1..2*n} j^a(2*n,k,j). Here the elemntary symmetric functions are e_j = e_j^{(N)} for N indeterminates x_1, ..., x_N, for any N >= 1.

A324247 Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n in terms of the elementary symmetric functions (using partitions in the Abramowitz-Stegun order).

Original entry on oeis.org

1, -1, 0, 1, -1, 0, -1, 1, 0, -1, 0, 1, -1, -1, 1, 1, -1, 0, -1, 1, 1, 0, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, -1, 1, 2, 1, 1, -1, -3, -1, 1, 2, -1, 0, -1, 1, 1, 1, 0, -1, -2, -1, -1, -1, 1, 2, -2, 3, 0, -1, -3, 0, 1, 2, -1, 0, 1, -1, -1, -1, -1, 1, 2, 2, 1, 1, 2, 0, -1, -3, -3, -3, -2, -1, 1, 4, 2, 5, 1, -1, -5, -3, 1, 3, -1, 0, -1, 1, 1, 1, 1, 0, -1, -2, -2, -1, -1, -1, -1, -1, 1, 3, 2, 1, 2, 5, 1, 1, 1, -1, -3, -3, -5, -5, -3, 0, 1, 4, 2, 8, 2, -1, -5, -4, 1, 3, -1, 0

Offset: 1

Wolfdieter Lang, May 23 2019

The length of row n is A000041(n).
The (one part) Witt symmetric function w_n is defined in the links below (one can add w_0 = 1). It can be expressed in terms of the elementary symmetric functions {e_i}{i=1..n} by using first a recurrence to express w_n in terms of the power sum symmetric functions p_n = Sum{1>=1} x_i^n, for the indeterminates {x_i}, by w_n = (1/n)*(p_n - Sum_{d|n, 1 <= d < n} d*(w_d)^{n/d}), n >= 2, with w_1 = p_1 = e_1. (See the array A324253). The p_n can then be expressed in terms of {e_i}_{i=1..n} by the Newton recurrence or its solution, the Girard-Waring formula (see A115131, row n, with partitions in the Abramowitz-Stegun order).
A relation between {w_n}{n>=1}, {e_i}{i>=0}, with e_0 = 1, and the indeterminates {x_i}{i>=1} is: Product{n>=0}(1 - w_n*t^n) = Sum_{i>=0} e_i*(-t)^i = Product_{j>=1} (1 - x_j*t). See the links.
If only N indeterminates {x_i}_{i=1..N} are considered all coefficients corresponding to partitions with at least one part > N are set to 0 (in addition to the ones given in the sequence).

			The irregular triangle (partition array) begins:
n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
------------------------------------------------------------------------------
1:   1
2:  -1  0
3:   1 -1  0
4:  -1  1  0 -1  0
5:   1 -1 -1  1  1 -1  0
6:  -1  1  1  0 -1 -1  0  1  1 -1  0
7:   1 -1 -1 -1  1  2  1  1 -1 -3 -1  1  2 -1  0
8:  -1  1  1  1  0 -1 -2 -1 -1 -1  1  2 -2  3  0 -1 -3  0  1  2 -1  0
...
n = 9: 1 -1 -1 -1 -1 1 2 2 1 1 2 0 -1 -3 -3 -3 -2 -1 1 4 2 5 1 -1 -5 -3 1 3 -1 0;
n = 10: -1 1 1 1 1 0 -1 -2 -2 -1 -1 -1 -1 -1 1 3 2 1 2 5 1 1 1 -1 -3 -3 -5 -5 -3 0 1 4 2 8 2 -1 -5 -4 1 3 -1 0;
...
---------------------------------------------------------------------------------
w_1 = e_1;
w_2 = - e_2 + 0;
w_3 = e_3 - e_1*e_2 + 0;
w_4:= - e_4 + e_1*e_3 + 0 - (e_1)^2*e_2 + 0;
w_5 = e_5 - e_1*e_4 - e_2*e_3 + (e_1)^2*e_3 + e_1*(e_2)^2 - (e_1)^3*e_2 + 0;
w_6 = - e_6 + e_1*e_5 + e_2*e_4 + 0 - (e_1)^2*e_4 - e_1*e_2*e_3 + 0 + (e_1)^3*e_3 + (e_1)^2*(e_2)^2 - (e_1)^4*e_2 + 0;
...
---------------------------------------------------------------------------------

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]
H J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015, Section 4.5., pp. 295-296 [with theta -> w, and the n = 1..6 results on p. 295]
SAGE, Witt symmetric functions

Cf. A000041, A115131 (Waring numbers), A324253 (with power sums).

w_n is given by the recurrence given in the comment above via the power sum symmetric functions {p_i} expressed in terms of the elementary symmetric functions {e_i}.

T(n, k) gives the coefficient of (e_1)^{a(k,1)}* ... *(e_n)^{a(k,n)} for w_n, corresponding to the k-th partition of n in Abramowitz-Stegun order, written as 1^{a(k,1)}* ... *n^{a(k,n)}, with nonnegative integers a(k,j) satisfying Sum_{j=1..n} j*a(k,j) = n, and the number of parts is Sum_{j=1..n} a(k,j) =: m.

A324253 Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n, multiplied by n!, in terms of the power sum symmetric functions (using partitions in the Abramowitz-Stegun order).

Original entry on oeis.org

1, 1, -1, -2, 0, -2, 6, 0, -3, 6, -9, 24, 0, 0, 0, 0, 0, -24, 120, 0, 0, -40, 0, 0, -30, 80, 90, -90, -130, 720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -720, 5040, 0, 0, 0, -1260, 0, 0, 0, 1260, 0, 0, 2520, 3780, 0, -945, 3780, 0, 0, 0, -6930, 6300, -8505

Offset: 1

Wolfdieter Lang, Jun 05 2019

The length of row n is A000041(n).
The (one part) Witt symmetric function w_n is defined in the links below. One should add w_0 = 1. It can be expressed in terms of the power sum symmetric functions p_k = Sum_{i>=1} (x_i)^k for the indeterminates {x_i}, by using  the recurrence w_n = (1/n)*(p_n - Sum_{d|n,1 <= d < n} d*(w_d)^{n/d}), n >= 2, with w_1 = p_1.
In order to have integer coefficients n!*w_n is considered, and terms are listed in the Abramowitz-Stegun order (with rising number of parts).
A logarithmic generating function of the power sums is related to the {w_n}{n>=1} sequence by Lp(t) := -Sum{j>=1} p_j*(t^j)/j = log(Product_{n>=0} (1 - w_n*t^n)). See the links.
If only N indeterminates {x_i}_{i=1..N} are considered all coefficients corresponding to partitions with at least one part > N are set to 0 (in addition to the ones given in the sequence).

			The irregular triangle (partition array) begins:
n\k    1  2  3   4  5  6   7   8   9  10   11 12 13 14  15  ...
---------------------------------------------------------------
1:     1
2:     1 -1
3:    -2  0 -2
4:     6  0 -3   6 -9
5:    24  0  0   0  0  0 -24
6:   120  0  0 -40  0  0 -30  80  90 -90 -130
7:   720  0  0   0  0  0   0   0   0   0    0  0  0  0 -720
...
n = 8:  5040 0 0 0 -1260 0 0 0 1260 0 0 2520 3780 0 -945 3780 0 0 0 -6930 6300 -8505;
n = 9: 40320 0 0 0 0 0 0 0 0 0 0 -4480 0 0 0 0 0 0 0 0 13440 0 0 0 0 0 -13440 0 0 -35840;
n = 10: 362880 0 0 0 0 -725760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -22680 145152 0 0 0 113400 0 0 -226800 0 226800 -113400 -412776;
...
---------------------------------------------------------------
w_1 = p_1;
w_2 = (1/2)*(p_2 - (p_1)^2);
w_3 = (1/3!)*(2*p_3 + 0 - 2*(p_1)^3);
w_4 = (1/4!)*(6*p_4 + 0 - 3*(p_2)^2 + 6*(p_1)^2*p_2 - 9*(p_1)^4);
w_5 = (1/5!)*(24*p_5 + 0 + 0 + 0 + 0 + 0 - 24*(p_1)^5) = (1/5)*(p_5 - (p_1)^5);
w_6 = (1/6!)*(120*p_6 + 0 + 0 - 40*(p_3)^2 + 0 + 0 - 30*(p_2)^3 + 80*(p_1)^3*p_3 + 90*(p_1)^2*(p_2)^2 - 90*(p_1)^4*p_2 - 130*(p_1)^6)
  = (1/72)*(12*p_6 - 4*(p_3)^2 - 3*(p_2)^3 + 8*(p_1)^3*p_3 + 9*(p_1)^2*(p_2)^2 -   9*(p_1)^4*p_2 - 13*(p_1)^6);
...
---------------------------------------------------------------

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]
H J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015, Section 4.5., pp. 295-296 [with theta -> w, psi-> p, and the n = 1..6 results on p. 295]
SAGE, Witt symmetric functions

Cf. A000041, A115131 (Waring numbers), A324247.

w_n is given by the recurrence given in the comment above in terms of the power sum symmetric functions {p_i}_{i>=1}, for n >= 1.

T(n, k) gives the coefficient of (p_1)^{a(1,k)}*...*(p_n)^{a(n,k)} for n!*w_n, corresponding to the k-th partition of n in Abramowitz-Stegun order, written as 1^{a(1,k)}*...*n^{a(n,k)}, with nonnegative integers a(n,j) satisfying Sum_{j=1..n} j*a(n,j) = n. The number of parts is Sum_{j=1..n} a(n,k) =: m(k).

A330415 Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 2, -1, 3, -3, 4, 1, -2, -4, 5, 4, 6, -5, -5, -1, 7, 5, 8, 5, -6, -6, 9, -5, -3, -7, 2, 6, 10, 12, 11, 1, -7, -8, -7, -9, 12, -9, -8, -6, 13, 14, 14, 7, 7, -10, 15, 6, -4, 7, -9, 8, 16, -7, -8, -7, -10, -11, 17, -21, 18, -12, 8, -1, -9, 16, 19, 9, -11, 16

Offset: 1

Gus Wiseman, Dec 14 2019

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330417.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

Table[If[n==1,0,(-1)^(PrimeOmega[n]-1)*Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])],{n,30}]

a(n) = (-1)^(Omega(n) - 1) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, -2, 1, 3, -3, -4, 1, 2, 4, 5, -4, -6, -5, -5, 1, 7, 5, -8, 5, 6, 6, 9, -5, 3, -7, -2, -6, -10, -12, 11, 1, -7, 8, -7, 9, -12, -9, 8, 6, 13, 14, -14, 7, 7, 10, 15, -6, 4, 7, -9, -8, -16, -7, 8, -7, 10, -11, 17, -21, -18, 12, -8, 1, -9, -16, 19, 9, -11

Offset: 1

Gus Wiseman, Dec 14 2019

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330415.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

Table[If[n==1,0,With[{tot=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]},(-1)^(tot-PrimeOmega[n])*tot*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])]],{n,30}]

a(n) = (-1)^(A056239(n) - Omega(n)) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

cp's OEIS Frontend

A308684 Partition array T(n, k) for the coefficients of the n-th power sums of the second elementary symmetric function in terms of the elementary symmetric functions.

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A324247 Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n in terms of the elementary symmetric functions (using partitions in the Abramowitz-Stegun order).

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A324253 Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n, multiplied by n!, in terms of the power sum symmetric functions (using partitions in the Abramowitz-Stegun order).

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A330415 Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

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A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

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