A321746 -id:A321746 - cp's OEIS frontend

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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

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).

Also the coefficient of h(v) in f(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321746.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321912-A321935.

A321747 Sum of coefficients of elementary symmetric functions in the monomial symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 1, -2, -1, 1, 1, 2, 1, -3, -1, -2, -2, 1, 1, 3, -1, 3, 2, 2, 1, -4, 1, -2, -1, -3, -1, -6, 1, 1, -2, 2, -2, 6, -1, -2, 2, 4, 1, 6, -1, 3, 3, 2, 1, -5, 1, 3, -2, -3, -1, -4, 2, -4, 2, -2, 1, -12, -1, 2, -3, 1, -2, -6, 1, 3, -2, -6, -1, 10, 1, -2

Offset: 1

Gus Wiseman, Nov 19 2018

sign

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

			The sum of coefficients of m(2211) = 9e(6) + e(42) - 4e(51) is a(36) = 6.

Wikipedia, Symmetric polynomial

Row sums of A321746. An unsigned version is A008480.

Cf. A005651, A056239, A124794, A124795, A296150, A321738, A321742-A321765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*Length[Permutations[primeMS[n]]],{n,50}]

a(n) = (-1)^(A056239(n) - A001222(n)) * A008480(n).

cp's OEIS Frontend

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs