A321908 -id:A321908 - cp's OEIS frontend

A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 2, -3, 1, 6, 3, 8, 6, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 2, -1, 0, -2, 1, -6, 3, 8, -6, 1, 24, 30, 20, 15, 20, 10, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5

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).

We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321900.

Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.

A321907 If n > 1 is the k-th prime number, then a(n) = k!, otherwise a(n) = 0.

Original entry on oeis.org

1, 1, 2, 0, 6, 0, 24, 0, 0, 0, 120, 0, 720, 0, 0, 0, 5040, 0, 40320, 0, 0, 0, 362880, 0, 0, 0, 0, 0, 3628800, 0, 39916800, 0, 0, 0, 0, 0, 479001600, 0, 0, 0, 6227020800, 0, 87178291200, 0, 0, 0, 1307674368000, 0, 0, 0, 0, 0, 20922789888000, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2018

nonn

1 is taken to be the zeroth prime number.

a(n) is the sum of coefficients of power sums symmetric functions in |y|! * s(y) / syt(y), where y is the integer partition with Heinz number n, s is Schur functions, and syt(y) is the number of standard Young tableaux of shape y.

Row sums of A321900.

Cf. A000085, A049084, A056239, A082733, A124795, A153452, A296150, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765, A321908.

Table[If[n==1,1,If[PrimeQ[n],PrimePi[n]!,0]],{n,40}]

a(n) = if (n==1, 1, if (isprime(n), primepi(n)!, 0)); \\ Michel Marcus, Nov 23 2018

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A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

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A321907 If n > 1 is the k-th prime number, then a(n) = k!, otherwise a(n) = 0.

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