A275787 Number of cells in the two-sided Coxeter complex of type B_n.

1, 5, 41, 509, 8469, 176217, 4400325, 128203049, 4268957449, 159922273421, 6656731517249, 304797275277365, 15224868078068845, 823874409422614577, 48012621942105876301, 2997884066292303095889, 199666128081901473290833, 14129411123649333432720277, 1058688691179737704258634521, 83732563305101190468369022317, 6971039973751002759723517967941

Offset: 1

Author

Kyle Petersen, Aug 09 2016

Keywords

• nonn

Comments

a(n) is the number of nonnegative integer matrices with sum of entries equal to 2*n-2 (or 2*n-1), no zero rows or columns, which are centrally symmetric. - Ludovic Schwob, Feb 17 2024

Examples

			The a(2) = 5 matrices whose sum of entries is equal to 2:
  [2] [1 1]
.
  [1] [1 0] [0 1]
  [1] [0 1] [1 0]
		

Links

• Evgeniy Krasko, Counting Unlabelled Chord Diagrams of Maximal Genus, arXiv:1709.00796 [math.CO], 2017.
• T. K. Petersen, A two-sided analogue of the Coxeter complex, arXiv:1607.00086 [math.CO], 2016.

Crossrefs

Cf. A120733 gives the number of cells for type A_n.

Programs

Maple

B:=proc(n) local f;
option remember;
if n=1 then 1+s*t;
elif n>1 then
f:=B(n-1);
RETURN(simplify( (2*n*s*t-s*t+1)*f+(2*s*t*(1-s)+s/n*(1-s)*(1-t))*diff(f,s) + (2*s*t*(1-t)+t/n*(1-s)*(1-t))*diff(f,t) + 2/n*s*t*(1-s)*(1-t)*diff( diff(f,s),t) ));
fi;
end:
seq(eval(eval(subs(s=x/(1+x),t=y/(1+y), B(n))*(1+x)^n*(1+y)^n,y=1),x=1), n=1..30);

Mathematica

B[n_] := B[n] = Which[n == 1, 1 + s*t, n > 1, f = B[n - 1]; Return[ Simplify[ (2*n*s*t - s*t + 1)*f + (2*s*t*(1 - s) + s/n*(1 - s)*(1 - t))*D[f, s] + (2*s*t*(1 - t) + t/n*(1 - s)*(1 - t))*D[f, t] + 2/n*s*t*(1 - s)*(1 - t)*D[ D[f, s], t]]]];
Join[{1}, Table[bn = ((B[n] /. {s -> x/(1 + x), t -> y/(1 + y)})*(1 + x)^n*(1 + y)^n /. {y -> 1, x -> 1}); Print[bn]; bn, {n, 1, 20}]] (* Jean-François Alcover, Nov 27 2017, from Maple *)