Formulas

Bruijn page 167 BLL page 109
$$P_G \left( \frac {\partial}{\partial x_1}, \frac {\partial}{\partial x_2}, \frac {\partial}{\partial x_2}, \cdots  \right)  P_H ( z_1, 2z_2, 3z_3, \cdots) $$

$$Z_F \left( \frac {\partial}{\partial x_1}, 2\frac {\partial}{\partial x_2}, 3\frac {\partial}{\partial x_2}, \cdots  \right) Z_G ( x_1, x_2, x_3, \cdots) $$

page 17, the cycle index
a) $$ Z_F ( \ x_1, x_2, x_3 ,...) = \sum_{n \ge 0} {1 \over n!} \left ( \sum_{\sigma \in S_n} fix F[\sigma] x_1^{\sigma_1}x_2^{\sigma_2} x_3^{\sigma_3} \dots \right)$$

page 18: exponential and type series from cycle index
a) $$ F(x) = Z_F ( \ x, 0, 0,...) $$

b) $$ \tilde F(x) = Z_F ( \ x \, x^2, x^3...) $$

page 29...43 Operations
page 29: Sum

$$ (F+G)[U] = \sum_{U=U} F[U]+G[U] $$

page 32: Product

$$ (F.G)[U] = \sum_{U_1+U_2=U} F[U_1] \times G[U_2] $$

page 38: Composition

$$ (F(G))[U] = \sum_{U_1+U_2+U_3+ \dots=U} F[\pi] \times G[U_1] \times G[U_2] \times G[U_3] \times \dots$$

page 43: Derivative

$$ F'[U] = \sum_{U'+ \star} F[U'] $$

page 39, composition series
a) $$ (F \circ G )(x) = F(G(x))\ $$

b) $$ \tilde {F \circ G}  = Z_F \left ( \ \tilde G(x), \ \tilde G(x^2), \ \tilde G(x^3), ... \right )  $$

c) $$ Z_{ F \circ G}(x_1, x_2, x_3,... ) = Z_F ( \ Z_G(x_1, x_2,... ), \ Z_G(x_2, x_4,... ),... )\ $$

page 90 - multisort
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a) $$ F(G,H) (x,y) = F(G(x,y), H(x,y) $$

b) $$ \tilde {F(G,H)} (x,y) = Z_F ( / \tilde G(x,y)), \tilde G(x^2, y^2),...; \tilde H(x,y), \tilde H(x^2, y^2),...) $$

c) $$ Z_{F(G,H)} = Z_F(Z_G,Z_H) = Z_F( Z_{G,1}, Z_{G,2},...; Z_{H,1}, Z_{H,2},... ) $$

where $$ Z_{G,k} (x_1, x_2, x_3,... ) = Z_G (x_k, x_{2k},x_{3k}, ...) $$

The Wreath Product
In group theory the composition of species coresponds to the wreath product.

For example, the table of Cyc3(Ens2) is :


 * $$(F \circ G)[A] = \sum_{\pi \in P[A]} (F[\pi] \times \Pi_{B \in \pi} G[B]).$$