Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #13 Sep 01 2020 18:42:08
%S 1,0,1,0,1,2,1,0,1,7,13,11,5,1,0,1,14,64,144,192,167,98,38,9,1,0,1,23,
%T 187,796,2134,3968,5400,5541,4347,2611,1188,399,94,14,1
%N Irregular triangle read by rows: T(n,k) = number of initially connected tournaments on n elements with k primary ascents.
%C This continues the enumeration of certain tournaments from A337271.
%C A tournament on {1,...,n} is a digraph with arcs between i and j for each i<j; either i->j or i<-j. The arc is an "ascent" if i->j.
%C It is a "primary ascent" if it's an ascent for which i and j belong to the same "initially connected component".
%C The initially connected components of a tournament on {1,...,n} are defined as follows: One component consists of everything reachable from 1. Remove that component; the next component, if any vertices remain, consists of everything reachable from the smallest remaining vertex. And so on.
%C The rows of this triangle are the coefficients of the polynomials g[n] (or g(n)) defined in the Mma code.
%F See Mma code.
%e Triangle begins:
%e 1,
%e 0,1,
%e 0,1,2,1,
%e 0,1,7,13,11,5,1,
%e 0,1,14,64,144,192,167,98,38,9,1,
%e 0,1,23,187,796,2134,3968,5400,5541,4347,2611,1188,399,94,14,1,
%e ...
%e For example, there are 7 initially connected tournaments on 4 elements with 2 ascents; they're the tournaments whose ascents are
%e 1->2,1->4; 1->2,2->4; 1->3,1->4; 1->3,2->4; 1->3,3->4; 1->4,2->3; 1->4,2->4.
%e Notice that this sequence has no row 0.
%t (* first create the Gaussian binomial coefficients {n\choose k}_q *)
%t gb[n_,k_]:=gb[n,k]=Expand[q^k gb[n-1,k]+gb[n-1,k-1]]
%t gb[n_,0]:=1
%t gb[0,k_]:=0
%t (* now define the gf for a single component *)
%t g[n_]:=g[n]=Expand[(1+q)^Binomial[n,2]-Sum[gb[n-1,k]g[n-k](1+q)^Binomial[k,2],{k,n-1}]]
%t (* now define the gf for a full tournament *)
%t h[n_]:=n!Coefficient[Series[Exp[Sum[g[k]z^k/k!,{k,n}]],{z,0,n}],z,n]
%t (* The elements of row n are the coefficients of g[n]. *)
%Y Cf. A337271.
%K nonn,tabf,more
%O 0,6
%A _N. J. A. Sloane_, Sep 01 2020, based on a communication from _Don Knuth_, Aug 25 2020