%I #37 Jul 08 2023 20:14:46
%S 1,1,4,34,488,10512,316224,12649104,649094752,41568338240,
%T 3249938294656,304670810708736,33736950933298688,4356802177994094080,
%U 649031480783423250432,110477935456564190447616,21310050396755400705088512,4623833701942527407032074240
%N Number of top simplices in a minimal triangulation of the permutohedron.
%C The analogous sequence with associahedron in place of permutohedron is (n+1)^{n-1}.
%C This also counts maximal chains in the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - _F. Chapoton_, Apr 29 2015
%C The ordinary generating series may be Gevrey of order 2, i.e., the coefficients may be bounded by A*B^n*(n!)^2 for some constants A and B. - _F. Chapoton_, Jul 07 2023
%H M. F. Hasler, <a href="/A111169/b111169.txt">Table of n, a(n) for n = 0..100</a>
%H Jean-Louis Loday, <a href="https://web.archive.org/web/20160803045214/http://www-irma.u-strasbg.fr/~loday">More information</a>
%H Jean-Louis Loday, <a href="http://arxiv.org/abs/math.AT/0510380">Parking functions and triangulation of the associahedron</a>, arXiv:math/0510380 [math.AT], 2005.
%H N. Reading, <a href="http://arxiv.org/abs/0909.3288">Noncrossing partitions and the shard intersection order</a>, arXiv:0909.3288 [math.CO], 2009.
%H N. Reading, <a href="http://dx.doi.org/10.1007/s10801-010-0255-3">Noncrossing partitions and the shard intersection order</a>, J. Algebraic Combin. 33 (2011), no. 4.
%F a(n) = Sum_{m=0..n-1} (binomial(n+1, m+1) - 1)*binomial(n-1, m)*a(m)*a(n-m-1). - _Robert G. Wilson v_, Oct 31 2005
%p function y=binom(n,p); y=1; for j = 0 : p-1; y=y*(n-j); end; for j = 1 : p; y=y/j; end; format long; nmax = 14; mm=nmax+1; zp=zeros(mm,1); zp(1:1) = 1; for n = 1 : nmax; z=0; for p = 0 : n-1; z=z+ (binom(n+1,p+1)-1) * binom(n-1,p) * zp(p+1:p+1) * zp(n-p:n-p); end; zp(n+1:n+1)=z; z; end; n, z
%t f[0] = 1; f[n_] := Sum[(Binomial[n + 1, m + 1] - 1)Binomial[n - 1, m]f[m]f[n - m - 1], {m, 0, n - 1}]; Table[f[n], {n, 0, 16}] (* _Robert G. Wilson v_, Oct 31 2005 *)
%o (PARI) a111169=[1]; A111169(n)={n<3&&return(n^n); global(a111169); while(n>m=#a111169, a111169=concat(a111169,sum(k=1,m-1,(binomial(m+2,k+1)-1)*binomial(m,k)*a111169[k]*a111169[m-k],2*(m+1)*a111169[m]))); a111169[n]} \\ _M. F. Hasler_, May 02 2015
%o (Sage)
%o @cached_function
%o def a(n):
%o if n == 0:
%o return 1
%o return sum((binomial(n + 1, m + 1) - 1) * binomial(n - 1, m)
%o * a(m) * a(n - m - 1) for m in range(n))
%o # _F. Chapoton_, Jul 07 2023
%K easy,nonn
%O 0,3
%A Jean-Louis Loday (loday(AT)math.u-strasbg.fr), Oct 21 2005
%E More terms from _Robert G. Wilson v_, Oct 31 2005