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A375504 Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.

%I A375504 #15 Sep 03 2024 01:01:17
%S A375504 1,1,1,2,3,1,6,12,6,1,24,62,39,10,1,120,396,296,95,15,1,720,3024,2616,
%T A375504 980,195,21,1,5040,26928,26568,11240,2605,357,28,1,40320,274320,
%U A375504 305892,143464,37290,5971,602,36,1,362880,3149280,3945024,2027460,578514,103824,12292,954,45,1
%N A375504 Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.
%C A375504 In a linear chord diagram a "bubble" is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. In a crystallized diagram, all chords are either short, or "bridge" two distinct bubbles, i.e., they have one vertex in one bubble, and the other vertex in a separate bubble. T(n,k) is the number of such crystallized diagrams built from n > 0 chords, exactly k > 0 of which are short.
%H A375504 Donovan Young, <a href="https://arxiv.org/abs/2408.17232">Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams</a>, arXiv:2408.17232 [math.CO], 2024.
%e A375504 Triangle begins:
%e A375504     1;
%e A375504     1,   1;
%e A375504     2,   3,   1;
%e A375504     6,  12,   6,  1;
%e A375504    24,  62,  39, 10,  1;
%e A375504   120, 396, 296, 95, 15, 1;
%e A375504   ...
%e A375504 For n = 3, let the vertices of the linear chord diagram be A,B,C,D,E,F. There are two diagrams with a single short chord: (AF)(BE)(CD) and (AE)(BF)(CD), and so T(3,1) = 2. There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE), and so T(3,2) = 3. Finally, there is one diagram with all three chords short: (AB)(CD)(EF), and so T(3,3)=1.
%t A375504 F[n_]:=Sum[Factorial2[2*i-1]*x^i,{i,0,n}];
%t A375504 T[n_,k_]:=Sum[(-1)^(n-k-l)*Factorial2[2*l-1]*Binomial[2*n-k,2*l]*Coefficient[F[n]^(k+1),x,n-k-l],{l,0,n-k}];
%Y A375504 Row sums give A375505.
%Y A375504 First column gives A000142.
%Y A375504 The second diagonal is A000217.
%K A375504 nonn,tabl
%O A375504 0,4
%A A375504 _Donovan Young_, Aug 18 2024