A359985 -id:A359985 - cp's OEIS frontend

A140945 Triangle read by rows: counts series-parallel networks by the number of series connections.

1, 1, 1, 1, 6, 1, 1, 25, 25, 1, 1, 90, 290, 90, 1, 1, 301, 2450, 2450, 301, 1, 1, 966, 17451, 41580, 17451, 966, 1, 1, 3025, 112035, 544971, 544971, 112035, 3025, 1, 1, 9330, 671980, 6076350, 12122502, 6076350, 671980, 9330, 1, 1, 28501, 3846700, 60738700, 217523922, 217523922, 60738700, 3846700, 28501, 1

Offset: 1

Brian Drake, Jul 24 2008

T(n,k) is the number of series-parallel matroids on [n+1] of rank k. - Andrew Howroyd, Mar 08 2023

			Triangle begins:
  1;
  1,   1;
  1,   6,     1;
  1,  25,    25,     1;
  1,  90,   290,    90,     1;
  1, 301,  2450,  2450,   301,   1;
  1, 966, 17451, 41580, 17451, 966, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50; first 17 rows from Brian Drake, Jul 24 2008)
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
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 A006351.
Second column is A000392.
Cf. A359985.

N:=6: 1/a*log(1+a*y)+1*log(1+b*y)/b-y=x: solve(%, y):series(%, x, N): simplify(%, symbolic): convert(%, polynom): subs(b=1, %): R:= [seq(i!*coeff(%, x, i), i=1..N-1)]: seq( seq(coeff(R[i], a, j), j=0..i-1), i=1..N-1);

T(n) = {[Vecrev(p) | p<-Vec(serlaplace(intformal(serreverse(log(1 + x*y + O(x*x^n))/y + log(1 + x + O(x*x^n)) - x))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }  \\ Andrew Howroyd, Mar 08 2023

E.g.f. is reversion of log(1+ax)/a+log(1+bx)/b-x.

Let f(x,t) = (1+x)*(1+x*t)/(1-x^2*t) and let D be the operator f(x,t)*d/dx. Then the n-th row polynomial equals (D^n)(f(x,t)) evaluated at x = 0. - Peter Bala, Sep 29 2011

A359986 Number of quasi series-parallel matroids on [n].

Original entry on oeis.org

1, 2, 5, 16, 67, 374, 2795, 27472, 343129, 5242178, 94775045, 1976279584, 46619596363, 1225918376726, 35526110971595, 1124201803439968, 38555828073355441, 1424165085382532162, 56356111920073052165, 2378150844047351691376, 106590791633357076443347

Offset: 0

Andrew Howroyd, Mar 08 2023

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..200
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.

Row sums of A359985.

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

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

A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 15, 1, 0, 0, 0, 0, 75, 1, 0, 0, 0, 0, 735, 280, 1, 0, 0, 0, 0, 0, 9345, 938, 1, 0, 0, 0, 0, 0, 76545, 77805, 2989, 1, 0, 0, 0, 0, 0, 0, 1865745, 536725, 9285, 1, 0, 0, 0, 0, 0, 0, 13835745, 27754650, 3334870, 28446, 1, 0

Offset: 1

Andrew Howroyd, Mar 09 2023

			Triangle begins:
  1;
  0, 0;
  0, 1,  0;
  0, 0,  1,   0;
  0, 0, 15,   1,     0;
  0, 0,  0,  75,     1,     0;
  0, 0,  0, 735,   280,     1,    0;
  0, 0,  0,   0,  9345,   938,    1, 0;
  0, 0,  0,   0, 76545, 77805, 2989, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..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 A007834.
Cf. A140945, A359985, A361353.
Cf. A034941, A361282.

\\ B gives A359985 as e.g.f.
B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {my(v=Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))); vector(#v, n, Vecrev(v[n]/y, n))}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }

E.g.f.: A(x,y) = log(1 + B(x,y)) where B(x,y) is the e.g.f. of A361353.
E.g.f.: A(x,y) = log(B(log(1 + x), y)/(1 + x)) where B(x,y) is the e.g.f. of A359985.
T(2*n+1, n+1) = A034941(n).
T(2*n, n+1) = A361282(n).

A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 5, 1, 0, 0, 15, 16, 1, 0, 0, 0, 175, 42, 1, 0, 0, 0, 735, 1225, 99, 1, 0, 0, 0, 0, 16065, 6769, 219, 1, 0, 0, 0, 0, 76545, 204400, 32830, 466, 1, 0, 0, 0, 0, 0, 2747745, 2001230, 147466, 968, 1, 0, 0, 0, 0, 0, 13835745, 56143395, 16813720, 632434, 1981, 1

Offset: 1

Andrew Howroyd, Mar 09 2023

See Table 2 in the Ferroni/Larson reference.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 0,  5,   1;
  0, 0, 15,  16,     1;
  0, 0,  0, 175,    42,      1;
  0, 0,  0, 735,  1225,     99,     1;
  0, 0,  0,   0, 16065,   6769,   219,   1;
  0, 0,  0,   0, 76545, 204400, 32830, 466, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..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 A361354.

Cf. A140945, A359985, A361355.

\\ B gives A359985 as e.g.f.
B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {[Vecrev(p/y) | p<-Vec(serlaplace(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x) - 1))]}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }

E.g.f.: A(x,y) = B(log(1 + x), y)/(1 + x) - 1 where B(x,y) is the e.g.f. of A359985.

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A140945 Triangle read by rows: counts series-parallel networks by the number of series connections.

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A359986 Number of quasi series-parallel matroids on [n].

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A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

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PARI

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A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula