A321897 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

1, 1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 6, 3, 8, 6, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 24, 30, 20, 15, 20, 10, 1, 0, 0, 0, 1, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 0, 1, 720, 840, 504, 420, 630

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• nonn
• tabf

Comments

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).

Examples

			Triangle begins:
    1
    1
    1    1
    0    1
    2    3    1
    0    1    1
    6    3    8    6    1
    0    0    1
    0    1    0    2    1
    0    0    2    3    1
   24   30   20   15   20   10    1
    0    0    0    1    1
  120   90  144   40   15   90  120   45   40   15    1
    0    6    0    3    8    6    1
    0    0    2    3    2    4    1
    0    0    0    0    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    0    0    1    0    2    1
For example, row 14 gives: 12h(41) = 6p(41) + 3p(221) + 8p(311) + 6p(2111) + p(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A321898.

Cf. A005651, A008480, A056239, A124794, A124795, A319193, A321742-A321765, A321896.