A318407 Triangle read by rows: T(n,k) is the number of Markov equivalence classes whose skeleton is the caterpillar graph on n nodes that contain precisely k immoralities.
0, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 3, 1, 1, 7, 8, 3, 3, 1, 8, 13, 6, 4, 1, 10, 23, 16, 13, 6, 1, 1, 11, 31, 29, 19, 10, 1, 1, 13, 46, 59, 46, 39, 13, 5, 1, 14, 57, 90, 75, 58, 23, 6, 1, 16, 77, 153, 158, 147, 97, 39, 15, 1, 1, 17, 91, 210, 248, 222, 155, 62, 21, 1
Offset: 0
Examples
The triangle T(n,k) begins: n\k| 0 1 2 3 4 5 6 7 8 9 -----+------------------------------------------------ 0 | 0 1 | 1 2 | 1 3 | 1 1 4 | 1 2 5 | 1 4 1 1 6 | 1 5 3 1 7 | 1 7 8 3 3 8 | 1 8 13 6 4 9 | 1 10 23 16 13 6 1 10 | 1 11 31 29 19 10 1 11 | 1 13 46 59 46 39 13 5 12 | 1 14 57 90 75 58 23 6 13 | 1 16 77 153 158 147 97 39 15 1 14 | 1 17 91 210 248 222 155 62 21 1
Links
- A. Radhakrishnan, L. Solus, and C. Uhler. Counting Markov equivalence classes for DAG models on trees, arXiv:1706.06091 [math.CO], 2017; Discrete Applied Mathematics 244 (2018): 170-185.
Programs
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Mathematica
W[0] = 0; W[1] = 1; W[2] = 1; W[3] = 1 + x; W[4] = 1 + 2x; W[n_] := W[n] = If[EvenQ[n], W[n-1] + x W[n-2], (x+2) W[n-2] + (x^3 - x^2 + x - 2) W[n-3] + (x^2 + 1) W[n-4]]; Join[{0}, Table[CoefficientList[W[n], x], {n, 0, 14}]] // Flatten (* Jean-François Alcover, Sep 17 2018 *) -
PARI
pol(n) = if (n==0, 0, if (n==1, 1, if (n==2, 1, if (n==3, 1 + x, if (n==4, 1 + 2*x, if (n%2, (x + 2)*pol(n-2) + (x^3 - x^2 + x-2)*pol(n-3) + (x^2 + 1)*pol(n-4), pol(n-1) + x*pol(n-2))))))); row(n) = Vecrev(pol(n)); \\ Michel Marcus, Sep 04 2018
Formula
A recursion whose n-th iteration is a polynomial with coefficient vector the n-th row of T(n,k):
W_0 = 0
W_1 = 1
W_2 = 1
W_3 = 1 + x
W_4 = 1 + 2*x
for n>4:
if n is even:
W_n = W_{n-1} + x*W_{n-2}
if n is odd:
W_n = (x + 2)*W_{n-2} + (x^3 - x^2 + x-2)*W_{n-3} + (x^2 + 1)*W_{n-4}
(see Theorem 4.3 of Radhakrishnan et al. for proof.)
Comments