Pensieve header: Sep 29: The Catalan numbers.

Topics (in no particular order). Whatever you may suggest; whatever comes to my mind; the Fibonacci numbers; the Catalan numbers; the Jones polynomial; a more efficient Jones algorithm; a riddle on spheres; Khovanov homology; Γ-calculus; the Hopf fibration; Hilbert’s 13th problem; non-commutative Gaussian elimination; free Lie algebras; the Baker-Campbell-Hausdorff formula; wacky numbers; an order 4 torus; the Schwarz Lantern; knot colourings; the Temperley-Lieb pairing; the dodecahedral link; sound experiments; barycentric subdivisions; a Peano curve; braid closures and Vogel’s algorithm; the insolubility of the quintic; phase portraits; the Mandelbrot set; shadows of the Cantor Aerogel; quilt plots; some image transformations; De Bruijn graphs; the Riemann series theorem; finite type invariants and the Willerton fish.

Killing Time

IntegerDigits[371]

((IntegerDigits[371])^3)

Total[((IntegerDigits[371])^3)]

crit[n_] := (n==Total[((IntegerDigits[n])^3)])

Select[Range[100,999],crit]

Select[Range[100,999],crit] // Length

The Catalan Numbers

Project Idea. Make an easily extendible, n-dependent “Catalan objects poster”, as explained on the blackboard.

Clear[ts]

ts[n_Integer]:=ts[Range[0,n+1]]

ts[4]

leon[_] := 7

leon[dror]

dror[n_]:=(n^2)

dror[7]

dror[Cos[x]]

hang[n_Integer]:=n^2;
hang[n_Symbol]:=n^3

hang[4]

hang[dror]

Union[{1,2,3},{4,5,6}]

vs={a,b,c,d,e,f}; k=4;

Table[vs[[j]], {j,k,Length[vs]}]

?Prepend

Prepend[Table[vs[[j]], {j,k,Length[vs]}],vs[[1]]]

Prepend[vs[[k;;]],vs[[1]]]

vs[[2;;k]]

l={a,b,c}; r={x,y,z};

Flatten[Table[
Table[
f[l[[i]],r[[j]]],
{j,1,Length[r]}
],
{i,1,Length[l]}
]]

Flatten[Table[
f[l[[i]],r[[j]]],
{i,1,Length[l]}, {j,1,Length[r]}
]]

Flatten[Table[
f[l[[i]],r[[j]]],
{i,Length[l]}, {j,Length[r]}
]]

Flatten[Table[f[t1,t2],{t1,l}, {t2,r}]]

ts[{_,_}]={ds[]};
ts[vs_List] := Module[{l,r,k,t1,t2,tds},
Union@@Table[
l=ts[Prepend[vs[[k;;]],vs[[1]]]];
r=ts[vs[[2;;k]]];
Flatten[Table[
tds=Join[t1,t2];
If[k>3, AppendTo[tds, d[vs[[2]],vs[[k]]]]];
If[k<Length[vs], AppendTo[tds, d[vs[[1]],vs[[k]]]]];
tds,
{t1,l}, {t2,r}
]],
{k,3,Length[vs]}
]
]

ts[{0,1}]

ts[{0,1,2}]

ts[{0,1,2,3}]

ts[2]

ts[3]

ts[4]

ts[4] // Length

ts[5] // Length

ts[6] // Length

ts[4] /. ds[ls___] -> Graphics[{ls}] /. d[i_,j_]:>Line[{i,j}] /. j_Integer:>{Cos[((2π j)/6)], Sin[((2π j)/6)]}

Further topics

Class photo; EIWL 5-8, a riddle on spheres, Etienne’s project.