A321915 - OEIS

%I #5 Nov 22 2018 18:17:51

%S 1,2,-1,-1,1,3,-3,1,-3,5,-2,1,-2,1,4,-2,-4,4,-1,-2,3,2,-4,1,-4,2,7,-7,

%T 2,4,-4,-7,10,-3,-1,1,2,-3,1,5,-5,-5,5,5,-5,1,-5,9,5,-7,-9,9,-2,-5,5,

%U 11,-11,-8,10,-2,5,-7,-11,14,10,-14,3,5,-9,-8,10,12

%N Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(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.

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

%C Also the coefficient of e(v) in f(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

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

%e Tetrangle begins:

%e   (1):  1

%e .

%e   (2):   2 -1

%e   (11): -1  1

%e .

%e   (3):    3 -3  1

%e   (21):  -3  5 -2

%e   (111):  1 -2  1

%e .

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

%e   (22):   -2  3  2 -4  1

%e   (31):   -4  2  7 -7  2

%e   (211):   4 -4 -7 10 -3

%e   (1111): -1  1  2 -3  1

%e .

%e   (5):      5 -5 -5  5  5 -5  1

%e   (41):    -5  9  5 -7 -9  9 -2

%e   (32):    -5  5 11 11 -8 10 -2

%e   (221):    5 -7 11 14 10 14  3

%e   (311):    5 -9 -8 10 12 13  3

%e   (2111):  -5  9 10 14 13 17 -4

%e   (11111):  1 -2 -2  3  3 -4  1

%e For example, row 14 gives: m(32) = -5h(5) + 11h(32) + 5h(41) - 11h(221) - 8h(311) + 10h(2111) - 2h(11111).

%Y This is a regrouping of the triangle A321748. Row sums are A155972.

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

%K sign,tabf

%O 1,2

%A _Gus Wiseman_, Nov 22 2018