A359985 Triangle read by rows: T(n,k) is the number of quasi series-parallel matroids on [n] with rank k, 0 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1365, 651, 63, 1, 1, 127, 2667, 10941, 10941, 2667, 127, 1, 1, 255, 10795, 82215, 156597, 82215, 10795, 255, 1, 1, 511, 43435, 589135, 1988007, 1988007, 589135, 43435, 511, 1

Offset: 0

Author

Andrew Howroyd, Mar 08 2023

Keywords

• nonn
• tabl

Comments

A quasi series-parallel matroid is a collection of series-parallel matroids. See the Ferroni/Larson reference for a precise definition.

The first six rows of this triangle are the same as A022166.

Examples

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1365,   651,   63,   1;
  1, 127, 2667, 10941, 10941, 2667, 127, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
• Nicholas Proudfoot, Yuan Xu, and Ben Young, On the enumeration of series-parallel matroids, arXiv:2406.04502 [math.CO], 2024.

Crossrefs

Row sums are A359986.

Columns k=0..2 are A000012, A000225, A006095.

Cf. A022166, A140945.

Programs

PARI

\\ Proposition 2.3, 2.8 in Ferroni/Larson, compare A140945.
T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp(x*(y+1) + y*intformal( serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }