This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Martin Rubey has authored 9 sequences.
There is a single graph of treewidth 3 on 4 vertices, which is the complete graph.
[sum(1 for g in graphs(n) if g.treewidth() == 3) for n in range(9)]
There is a single graph of treewidth 2 on 3 vertices, which is the complete graph.
[sum(1 for g in graphs(n) if g.treewidth() == 2) for n in range(9)]
For n=5, A000112(n) - a(n) = 63 - 61 = 2 because we have 2 posets with 5 elements that are not ranked: a<b<c<d a<e<d and a<c<e a<d b<d b<e where < means "is covered by". - _Geoffrey Critzer_, Oct 29 2023
\\ See PARI link in A361953 for program code. A361920seq(20) \\ Andrew Howroyd, Apr 01 2023
sum(1 for P in posets(n) if P.is_ranked())
sum(1 for P in posets(n-2) if (Q := P.with_bounds()).is_graded() and Q.is_eulerian())
\\ See PARI link in A361957 for program code. A361912seq(20) \\ Andrew Howroyd, Apr 03 2023
sum(1 for P in posets(n) if P.is_graded())
xray[perm_List] := Module[{P, n = Length[perm]}, P[, ] = 0; Thread[perm -> Range[n]] /. Rule[i_, j_] :> Set[P[i, j], 1]; Table[Sum[P[i - j + 1, j], {j, Max[1, i - n + 1], Min[i, n]}], {i, 1, 2n - 1}]];
a[n_] := xray /@ Permutations[Range[n]] // Tally // Count[#, {_List, 1}]&;
Do[Print[n, " ", a[n]], {n, 0, 10}] (* Jean-François Alcover, Feb 28 2020 *)
def X_ray(pi):
P = Permutation(pi).to_matrix()
n = P.nrows()
return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n)))
for k in range(1,2*n))
@cached_function
def X_rays(n):
return sorted(X_ray(pi) for pi in Permutations(n))
def statistic(pi): return X_rays(pi.size()).count(X_ray(pi))
[[statistic(pi) for pi in Permutations(n)].count(1) for n in range(7)]
Length(ConjugacyClassesSubgroups(CoxeterGroup( "B", n )));
[#Subgroups(CoxeterGroup("B" cat IntegerToString(n))) : n in [1..9]]; // Robin Visser, Aug 09 2023
For n=5 there are 5 Ptolemy diagrams up to rotation: the pentagon with no diagonal, the pentagon with one diagonal, the pentagon with two noncrossing diagonals, the pentagon with three diagonals and the pentagon with all five diagonals.
seq(n)={my(p=serreverse(x - x^2*(1 + x)/(1 - x) + O(x*x^n)), P(d)=subst(p + O(x^(n\d+1)), x, x^d)); Vec(2*sum(d=1, n, eulerphi(d)/d*log(1/(1-P(d)))) - (P(1)^3 + 2*P(3))/3 - (3*P(1)^2+P(2))/2 - (2 - x)*P(1) - x^2)} \\ Andrew Howroyd, May 09 2023
For n=4 there are 4 Ptolemy diagrams: the square with no diagonal, two diagrams with one diagonal, and the square with both diagonals .
a(n) = n-=3; sum(i=0, floor((n+1)/2), 2^i*binomial(n+1+i,i)*binomial(2*n+2,n+1-2*i))/(n+2); \\ Michel Marcus, Jan 14 2012; corrected Jun 13 2022
seq(n) = Vec(x*(serreverse(x - x^2*(1 + x)/(1 - x) + O(x^(n+2))) - x)) \\ Andrew Howroyd, May 09 2023
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