Living Catalogue of Reversive Symbols

# Reversive Symbols for Rational Sequences

### Len Smiley, Univ. of Alaska Anchorage

 Still choosing layout, check back soon. 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 q-gon-free ``` F-2F2+Fq-1-Fq ------------ 1-F ``` Schroeder dissection, no tiles are q-gons Schroeder dissection, all tiles have more than q sides ``` F-F2-Fq ------------ 1-F ``` 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

Leonard Smiley
Created: Wed Jul 7 14:41:05 AKDT 1999 Last modified: Fri Jul 9 10:24:14 AKDT 1999