Table 5 Formulae for the expected numbers of triplets after three events

{t1,t1,t1}

-

u^′(1+v)³(1+w)³

-

{t1,t1,t2}

2uv(1+v)(1+w)³

2(3u^′+u)v(1+v)(1+w)³

2u(3v^′+v)

×(1+2v^′+v)(1+w)³

{t1,t2,t2}

-

-

-

{t2,t2,t2}

-

-

(1+u)v^′(1+w)³

{t2;t2;t3}

2(1+u)vw(1+w)

2(1+2u^′+u)vw(1+w)

2(1+u)(3v^′+v)w(1+w)

{t2,t3,t3}

-

-

-

{t3,t3,t3}

-

-

-

{t1,t1,t3}

2u(1+v)²w(1+w)

2(3u^′+u)(1+v)²w(1+w)

2u(1+2v^′+v)²w(1+w)

{t1,t3,t3}

-

-

-

{t1,t2,t3}

-

-

-

model \(M\in \mathscr {M}\)

triplet \(\Delta \in \mathscr {T}\)

(2,2,3)

(2,3,3)

(3,2,3)

{t1,t1,t1}

-

-

u^′(1+v)³

×(1+2w^′+w)³

{t1,t1,t2}

2uv(1+v)

2u(3v^′+v)(1+2v^′+v)

2(3u^′+u)v(1+v)

×(1+2w^′+w)³

×(1+2w^′+w)³

×(1+2w^′+w)³

{t1,t2,t2}

-

-

-

{t2,t2,t2}

-

(1+u)v^′

-

×(1+2w^′+w)³

{t2;t2;t3}

2(1+u)v

2(1+u)(3v^′+v)

2(1+2u^′+u)v

×(3w^′+w)(1+2w^′+w)

×(3w^′+w)(1+2w^′+w)

×(3w^′+w)(1+2w^′+w)

{t2,t3,t3}

-

-

-

{t3,t3,t3}

(1+u)(1+v)w^′

(1+u)(1+2v^′+v)w^′

(1+2u^′+u)(1+v)w^′

{t1,t1,t3}

2u(1+v)²

2u(1+2v^′+v)²

2(3u^′+u)(1+v)²

×(3w^′+w)(1+2w^′+w)

×(3w^′+w)(1+2w^′+w)

×(3w^′+w)(1+2w^′+w)

{t1,t3,t3}

-

-

-

{t1,t2,t3}

-

-

-

model \(M\in \mathscr {M}\)

triplet \(\Delta \in \mathscr {T}\)

(3,3,2)

(3,3,3)

{t1,t1,t1}

u^′(1+2v^′+v)³(1+w)³

u^′(1+2v^′+v)³(1+2w^′+w)³

{t1,t1,t2}

2(3u^′+u)(3v^′+v)(1+2v^′+v)(1+w)³

2(3u^′+u)(3v^′+v)(1+2v^′+v)(1+2w^′+w)³

{t1,t2,t2}

-

-

{t2,t2,t2}

(1+2u^′+u)v^′(1+w)³

(1+2u^′+u)v^′(1+2w^′+w)³

{t2;t2;t3}

2(1+2u^′+u)(3v^′+v)w(1+w)

2(1+2u^′+u)(3v^′+v)(3w^′+w)(1+2w^′+w)

{t2,t3,t3}

-

-

{t3,t3,t3}

-

(1+2u^′+u)(1+2v^′+v)w^′

{t1,t1,t3}

2(3u^′+u)(1+2v^′+v)²w(1+w)

2(3u^′+u)(1+2v^′+v)²(3w^′+w)(1+2w^′+w)

{t1,t3,t3}

-

-

{t1,t2,t3}

-

-