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Pray for MathML - Len |
| Catalan Numbers | F-F2 |
(1/(n+1))*binomial(2*n,n) |
Schroeder dissection, all tiles are triangles |
A000045 |
| 4-gon Numbers | F-F3 |
(1/(2*n+1))*binomial(3*n,n) |
Schroeder dissection, all tiles are quadrangles |
A001764 |
| 3- or 4-gon Numbers | F-F2-F3 |
sum(binomial(n+k,k)*binomial(k,n-k),k=ceil(n/2)..n)/(n+1); |
Schroeder dissection, all tiles are triangles or quadrangles |
A001002 |
| (q+1)-gon Numbers | F-Fq |
(1/((q-1)*n+1))*binomial(q*n,n) |
Schroeder dissection, all tiles are (q+1)-gons |
A002293 |
| Schroeder Numbers | F-2F2 -------- 1-F |
(1/(n+1))*sum(binomial(2*n-k,n)*binomial(n-1,k),k=0..n-1) |
Schroeder dissection |
A001003 |
| Double Schroeder Numbers | F-F2 -------- 1+F |
(2/(n+1))*sum(binomial(2*n-k,n)*binomial(n-1,k),k=0..n-1) |
Schroeder dissection, no diagonals to vertices 1 or 2 |
A006318 |
| Triangle-free | F-F2-F3 -------- 1-F |
(1/(n+1))*sum(binomial(n+k,k)*binomial(n-k-1,k-1),k=0..ceil((n-1)/2)) |
Schroeder dissection, no tiles are triangles |
A046736 |
| Quad-free | F-2F2+F3-F4 ------------ 1-F |
Schroeder dissection, no tiles are quadrangles |
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| q-gon-free | F-2F2+Fq-1-Fq ------------ 1-F |
Schroeder dissection, no tiles are q-gons |
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Schroeder dissection, all tiles have more than q sides |
F-F2-Fq ------------ 1-F |
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| Only odd-sided tiles | F-F2-F3 --------- 1-F2 |
(1/(n+1))*sum(binomial(2*n-2*k,n)*binomial(n-k-1,k),k=0..ceil((n+1)/2)) |
Schroeder dissection, all tiles have odd number of sides |
A049124 |
| Only even-sided tiles | F-2F3 --------- 1-F2 |
(1/(2*m+1))*sum(binomial(2*m+k,k)*binomial(m-1,k-1),k=0..m) |
Schroeder dissection, all tiles have even number of sides |
A003168 |
| Motzkin Numbers | F-F2 -------- 1-F3 |
(1/(n+1))*sum(binomial(n+1,k)*binomial(k,2*k-n-2),k=0..n+1) |
Ways to place non-touching chords with n labelled points as endpoints |
A001006 |
| "Motzkin sums" | F-F2 -------- 1-F+F2 |
(1/(n+1))*sum(binomial(n+1,k)*binomial(n-k-1,k-1),k=0..ceil((n)/2)) |
A005043 |