A375504 Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.
1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 62, 39, 10, 1, 120, 396, 296, 95, 15, 1, 720, 3024, 2616, 980, 195, 21, 1, 5040, 26928, 26568, 11240, 2605, 357, 28, 1, 40320, 274320, 305892, 143464, 37290, 5971, 602, 36, 1, 362880, 3149280, 3945024, 2027460, 578514, 103824, 12292, 954, 45, 1
Offset: 0
Examples
Triangle begins:
1;
1, 1;
2, 3, 1;
6, 12, 6, 1;
24, 62, 39, 10, 1;
120, 396, 296, 95, 15, 1;
...
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.
Links
- Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.
Programs
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Mathematica
F[n_]:=Sum[Factorial2[2*i-1]*x^i,{i,0,n}]; 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}];
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