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Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.
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%I #26 Jul 31 2024 11:24:41

%S 1,1,1,1,2,1,4,5,6,5,6,16,21,26,24,19,48,88,119,147,133,49,164,330,

%T 538,735,892,846,150,559,1302,2310,3568,4830,5876,5661,442,1952,5005,

%U 9882,16500,24596,33253,40490,39556,1424,6872,19504,41715,75387,120582,176354,237336,290020,286000,4522

%N Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.

%H W. G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]. See Table 1 (with a typo at G(n=1,m=6)).

%H L. March and C. F. Earl, <a href="https://doi.org/10.1068/b040057">On Counting Architectural Plans</a>, Environment and Planning B, 4 (1977), 57-80. See Table 2.

%H Jean-François Alcover, <a href="/A262586/a262586.txt">Mathematica code</a>

%F Brown (Eq. 6.3) gives a formula.

%e The first few rows are:

%e 1, 1, 1, 4, 6, 19, 49, 150, 442, 1424, 4522, 14924, 49536, ...

%e 1, 2, 5, 16, 48, 164, 559, 1952, ...

%e 1, 6, 21, 88, 330, 1302, 5005, 19504, 75582, 294140, ...

%e 5, 26, 119, 538, 2310, 9882, 41715, 175088, 730626, ...

%e ...

%e The first few antidiagonals are:

%e 1,

%e 1,1,

%e 1,2,1,

%e 4,5,6,5,

%e 6,16,21,26,24,

%e 19,48,88,119,147,133,

%e 49,164,330,538,735,892,846,

%e ...

%p A262586 := proc(n,m)

%p BrownG(n,m) ; # procedure in A210696

%p end proc:

%p for d from 0 to 12 do

%p for n from 0 to d do

%p printf("%d,",A262586(n,d-n)) ;

%p end do:

%p end do: # _R. J. Mathar_, Oct 21 2015

%t See LINKS section.

%Y Rows and columns include A002709, A002710, A002711, A001683, A210696, A005498, A005499.

%Y Antidiagonal sums are A341855.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Oct 20 2015