\psline(0,2)(1,2)
  \psline(1,3)(2,1)
  \endpspicture \qquad
  \pspicture(0,1)(2,3)
  \psdots(0,1)(0,2)(1,1)(1,2)(1,3)(2,1)(2,2)
  \psline(0,1)(1,1)
  \psline(0,2)(1,3)
  \psline(1,2)(2,1)
  \endpspicture \qquad
  \pspicture(0,1)(2,3)
  \psdots(0,1)(0,2)(1,1)(1,2)(1,3)(2,1)(2,2)
  \psline(0,1)(1,2)
  \psline(0,2)(1,3)
  \psline(1,1)(2,1)
  \endpspicture\end{aligned}$$ which give the three tuples of partial permutation matrices $$\begin{aligned}
  \left(\begin{pmatrix}
  1&0&0\\ 0&1&0
  \end{pmatrix},\begin{pmatrix}
    0&0 \\ 0&0 \\ 1&0
  \end{pmatrix}\right)\qquad
  \left(\begin{pmatrix}
  1&0&0\\ 0&0&1
  \end{pmatrix},\begin{pmatrix}
    0&0 \\ 1&0 \\ 0&0
  \end{pmatrix}\right)\qquad
  \left(\begin{pmatrix}
  0&1&0\\ 0&0&1
  \end{pmatrix},\begin{pmatrix}
    1&0 \\ 0&0 \\ 0&0
  \end{pmatrix}\right).\end{aligned}$$ The formula is then $$\begin{aligned}
  [\Omega] &=
  {\mathfrak{S}}_{123}(a;b){\mathfrak{S}}_{3412}(b;c)+{\mathfrak{S}}_{132}(a;b){\mathfrak{S}}_{3142}(b;c)+{\mathfrak{S}}_{231}(a;b){\mathfrak{S}}_{1342}(b;c)
  \\
  &= [(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2)] \\
  &+ [(a_1+a_2-b_1-b_2)((b_1-c_1)(b_1-c_2)(b_2+b_3-c_1-c_2))] \\
  &+ [(a_1-b_1)(a_2-b_1))((b_2-c_1)(b_3-c_1)+(b_1-c_2)(b_3-c_1)+(b_1-c_2)(b_2-c_2))]\end{aligned}$$ using, say, the pipe dream formula [@FK] [@KMS Thm. 5.3] to evaluate the double Schubert polynomials ${\mathfrak{S}}_w(x;y)$.

The multidegrees of quiver loci are particularly important for studying the singularities of composites of differential mappings (see [@BF; @FR; @BFR] and the references therein).

Until now, the only formulae for these multidegrees were in type $A$. The first such formula was in [@BF], and applied only to the case that the directed arrows are all oriented the same direction. This type $A$ formula has been improved in three ways: it has been made manifestly positive in an appropriate sense, the $K$-polynomial has been computed [@KMS], and the orientation has been generalized [@BR]. Some of these have been combined: the $K$-polynomial has been computed positively [@Bu; @Mi], and the 