A319191 -id:A319191 - cp's OEIS frontend

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

1, -1, 1, 2, -3, 1, -6, 3, 8, -6, 1, 24, -30, -20, 15, 20, -10, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 720, -840, -504, -420, 630, 504, 210, 280, -105, -210, -420, 105, 70, -21, 1, -5040, 5760, 3360, 1260, -3360, 2688, -1260, -4032, -3360, -1120

Offset: 1

Gus Wiseman, Sep 13 2018

A generalization of the triangle of Stirling numbers of the first kind, these are the coefficients appearing in the expansion of single-part augmented elementary symmetric functions in terms of power-sum symmetric functions.

			Triangle begins:
   1
  -1   1
   2  -3   1
  -6   3   8  -6   1
  24 -30 -20  15  20 -10   1
The fourth row corresponds to the symmetric function identity: 24 e(4) = -6 p(4) + 3 p(22) + 8 p(31) - 6 p(211) + p(1111).

Other row orderings are A036039 and A102189.

Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

A321748 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 4, -2, -4, 4, -1, 1, -2, 1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 5, -5, -5, 5, 5, -5, 1, 4, -4, -7, 10, -3, 6, -6, -6, -3, 2, 6, 12, -9, -6, 6, -1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, -1, 1, 2, -3, 1, 7

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the coefficient of e(v) in f(u), where e is elementary symmetric functions and f is forgotten symmetric functions.

			Triangle begins:
   1
   1
   2  -1
  -1   1
   3  -3   1
  -3   5  -2
   4  -2  -4   4  -1
   1  -2   1
  -2   3   2  -4   1
  -4   2   7  -7   2
   5  -5  -5   5   5  -5   1
   4  -4  -7  10  -3
   6  -6  -6  -3   2   6  12  -9  -6   6  -1
  -5   9   5  -7  -9   9  -2
  -5   5  11 -11  -8  10  -2
  -1   1   2  -3   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   5  -7 -11  14  10 -14   3
For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).

Wikipedia, Symmetric polynomial

Row sums are A080339.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319191, A319193, A321738, A321742-A321765.

A321749 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in h(u) or, equivalently, the coefficient of h(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
  -1   1
   0   1
   1  -2   1
   0  -1   1
  -1   1   2  -3   1
   0   0   1
   0   1   0  -2   1
   0   0   1  -2   1
   1  -2  -2   3   3  -4   1
   0   0   0  -1   1
  -1   2   2   1  -1  -3  -6   6   4  -5   1
   0  -1   0   1   2  -3   1
   0   0  -1   2   1  -3   1
   0   0   0   0   1
   1  -2  -2  -2   6   3   3   3  -4  -4 -12  10   5  -6   1
   0   0   0   1   0  -2   1
For example, row 14 gives: h(41) = -e(41) + e(221) + 2e(311) - 3e(2111) + e(11111).

Wikipedia, Symmetric polynomial

Row sums are A036987.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A319191, A319193, A321742-A321765.

A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 3, 1, -2, -1, 10, 3, 2, 1, -7, -1, -2, -2, 47, 1, 6

Offset: 1

Gus Wiseman, Nov 20 2018

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

			The sum of coefficients of p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211) is a(12) = -7.

Wikipedia, Symmetric polynomial

Row sums of A321888. An unsigned version is A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A319193, A321742-A321765.

A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 6, 1, 1, 1, 1, 1, 1, 3, 2, 4, 6, 1, 2, 2, 3, 4, 1, 4, 3, 7, 12, 1, 6, 4, 12, 24, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 1, 2, 3, 5, 4, 7, 10, 1, 3, 5, 11, 8, 18, 30, 1, 3, 4, 8, 7, 13, 20, 1, 4, 7, 18, 13, 33, 60, 1, 5

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 f(v) in e(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

			Tetrangle begins:
  (1):  1
.
  (2):   1  1
  (11):  1  2
.
  (3):    1  1  1
  (21):   1  2  3
  (111):  1  3  6
.
  (4):     1  1  1  1  1
  (22):    1  3  2  4  6
  (31):    1  2  2  3  4
  (211):   1  4  3  7 12
  (1111):  1  6  4 12 24
.
  (5):      1  1  1  1  1  1  1
  (41):     1  2  2  3  3  4  5
  (32):     1  2  3  5  4  7 10
  (221):    1  3  5 11  8 18 30
  (311):    1  3  4  8  7 13 20
  (2111):   1  4  7 18 13 33 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: h(32) = m(5) + 3m(32) + 2m(41) + 5m(221) + 4m(311) + 7m(2111) + 10m(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321744.

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

A321917 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 3, 6, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 2, 2, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, 1, 3, 4, 6, 6, 6, 0, 1, 5, 10, 30

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):   1
  (11):  1  2
.
  (3):    1
  (21):   1  1
  (111):  1  3  6
.
  (4):     1
  (22):    1  2
  (31):    1     1
  (211):   1  2  2  2
  (1111):  1  6  4 12 24
.
  (5):      1
  (41):     1  1
  (32):     1     1
  (221):    1  1  2  2
  (311):    1  2  1     2
  (2111):   1  3  4  6  6  6
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: p(32) = m(5) + m(32).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321750.

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

A321924 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in s(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 2, 3, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 4, 0, 0, 1, 2, 1, 3, 5, 0, 0, 0, 1, 0, 2, 5, 0, 0, 0, 1, 1, 3, 6, 0, 0, 0, 0, 0, 1, 4, 0, 0, 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).

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321761.

Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321925 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in m(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and m is monomial symmetric functions.

Original entry on oeis.org

1, 1, -1, 0, 1, 1, -1, 1, 0, 1, -2, 0, 0, 1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 1, 0, -1, 1, -1, 2, 0, 0, 0, 1, -3, 0, 0, 0, 0, 1, 1, -1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, 0, 0, 1, -1, -1, 2, -2, 0, 0, 0, 1, 0, -2, 3, 0, 0, 0, -1, 1, -1, 3, 0, 0, 0, 0, 0, 1

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):   1 -1
  (11):     1
.
  (3):    1 -1  1
  (21):      1 -2
  (111):        1
.
  (4):     1    -1  1 -1
  (22):       1    -1  1
  (31):      -1  1 -1  2
  (211):            1 -3
  (1111):              1
.
  (5):      1 -1        1 -1  1
  (41):        1 -1  1 -1  1 -2
  (32):           1 -1 -1  2 -2
  (221):             1    -2  3
  (311):            -1  1 -1  3
  (2111):                  1 -4
  (11111):                    1
For example, row 14 gives: m(32) = s(32) - s(221) - s(311) + 2s(2111) - 2s(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321763.

Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 2, 3, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, -2, -1, -2, -1, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 0, -6, -6, -5, -3, -3, -1, 0

Offset: 1

Gus Wiseman, Nov 23 2018

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

The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -1 -1
  (111):  2  3  1
.
  (4):    -1
  (22):    1  1
  (31):    1     1
  (211):  -2 -1 -2 -1
  (1111):  6  3  8  6  1
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    2  1  2  1
  (311):    2  2  1     1
  (2111):  -6 -6 -5 -3 -3 -1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: F(32) = -p(5) - p(32).

Wikipedia, Symmetric polynomial

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.

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

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.

cp's OEIS Frontend

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

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A321748 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

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A321749 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in h(u) or, equivalently, the coefficient of h(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

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A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

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A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

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A321917 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

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A321924 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in s(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

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A321925 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in m(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and m is monomial symmetric functions.

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A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

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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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