A321754 -id:A321754 - cp's OEIS frontend

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.

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.

A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 2, -1, 0, 1, 3, -3, 1, 0, 2, -1, 0, 0, 1, 4, -2, -4, 4, -1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1, 5, -5, -5, 5, 5, -5, 1, 0, 4, 0, -2, -4, 4, -1, 0, 0, 6, -6, -3, 5, -1, 0, 0, 0, 4, 0, -4, 1, 0, 0, 0, 0, 3, -3, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   2 -1
  (11):     1
.
  (3):    3 -3  1
  (21):      2 -1
  (111):        1
.
  (4):     4 -2 -4  4 -1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):            2 -1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):        4    -2 -4  4 -1
  (32):           6 -6 -3  5 -1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                  2 -1
  (11111):                    1
For example, row 14 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321754.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

cp's OEIS Frontend

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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Formula

A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

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