A359986 -id:A359986 - cp's OEIS frontend

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

Andrew Howroyd, Mar 08 2023

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.

			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;
  ...

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.

Row sums are A359986.
Columns k=0..2 are A000012, A000225, A006095.
Cf. A022166, A140945.

\\ 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])) }

A007834 Number of point labeled reduced 5-free two-graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 16, 76, 1016, 10284, 157340, 2411756, 44953712, 899824256, 20283419872, 495216726096, 13202082981712, 378896535199888, 11690436112988224, 385173160930360192, 13509981115738946816, 502374681770910293568, 19746124320077115154112, 817908018939079281840320

Offset: 1

Peter J. Cameron

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Row sums are A361355.

Cf. A007831, A007832, A007833, A359986.

CoefficientList[Series[-2*LambertW[-1/2*E^(-1/2)*(1+x)^(1/2)]/(1+x), {x, 0, 15}], x]* Range[0, 15]! (* Vaclav Kotesovec, Sep 30 2013 *)

\\ B(x) gives the e.g.f. of A359986.
B(n)={exp(2*x + intformal(serreverse(log(1 + x + O(x^n)) + log(1 + x + O(x^n)) - x)))}
seq(n)={Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))} \\ Andrew Howroyd, Oct 15 2024

E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))/(1+x). - Vladeta Jovovic, Aug 21 2006
a(n) ~ sqrt(2)*sqrt(4-exp(1)) * n^(n-1) / (8*exp(n-1)*(4*exp(-1)-1)^n). - Vaclav Kotesovec, Sep 30 2013
E.g.f.: log(B(log(1 + x))/(1 + x)), where B(x) is the e.g.f. of A359986. - Andrew Howroyd, Oct 15 2024

a(20) onwards from Andrew Howroyd, Oct 15 2024

A361354 Number of simple quasi series-parallel matroids on [n].

Original entry on oeis.org

1, 1, 2, 6, 32, 218, 2060, 23054, 314242, 4897410, 87427276, 1741312444, 38482278928, 931618115860, 24554678866736, 699328394272236, 21410158708401980, 701011980397033052, 24445424273647475096, 904440666571331841992, 35386719095200164370912, 1459756349974815778252152

Offset: 1

Andrew Howroyd, Mar 09 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.

Row sums of A361353.

Cf. A007834, A359986.

seq(n) = Vec(serlaplace( -1 + subst(exp(2*x + intformal(-x + 2*serreverse(1 + 2*x - exp(x + O(x^n))))), x, log(1 + x + O(x*x^n)))/(1 + x) ))

E.g.f.: B(1 + log(x))/(1 + x) - 1 where B(x) is the e.g.f. of A359986.

E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A007834.

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

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A007834 Number of point labeled reduced 5-free two-graphs with n nodes.

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A361354 Number of simple quasi series-parallel matroids on [n].

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