A321916 - OEIS

%I #4 Nov 22 2018 18:17:58

%S 1,-1,1,0,1,1,-2,1,0,-1,1,0,0,1,-1,1,2,-3,1,0,1,0,-2,1,0,0,1,-2,1,0,0,

%T 0,-1,1,0,0,0,0,1,1,-2,-2,3,3,-4,1,0,-1,0,1,2,-3,1,0,0,-1,2,1,-3,1,0,

%U 0,0,1,0,-2,1,0,0,0,0,1,-2,1,0,0,0,0,0,-1,1

%N Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

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

%C Also the coefficient of h(v) in e(u).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>

%e Tetrangle begins (zeroes not shown):

%e   (1):  1

%e .

%e   (2):  -1  1

%e   (11):     1

%e .

%e   (3):    1 -2  1

%e   (21):     -1  1

%e   (111):        1

%e .

%e   (4):    -1  1  2 -3  1

%e   (22):       1    -2  1

%e   (31):          1 -2  1

%e   (211):           -1  1

%e   (1111):              1

%e .

%e   (5):      1 -2 -2  3  3 -4  1

%e   (41):       -1     1  2 -3  1

%e   (32):          -1  2  1 -3  1

%e   (221):             1    -2  1

%e   (311):                1 -2  1

%e   (2111):                 -1  1

%e   (11111):                    1

%e For example, row 14 gives: h(32) = -e(32) + 2e(221) + e(311) - 3e(2111) + e(11111).

%Y This is a regrouping of the triangle A321749. Row sums are A134286.

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

%K sign,tabf

%O 1,7

%A _Gus Wiseman_, Nov 22 2018