OFFSET
1,2
COMMENTS
A Gog triangle of size n is a Gelfand-Tsetlin pattern (X_{i,j})_{1<=j<=i<=n} with bottom row 1 2 3 ... n and strictly increasing rows.
A Gelfand-Tsetlin pattern is said to be gapless if X_{i+1,j+1} - X_{i,j} is at most 1.
Gog triangles are in bijection with alternating sign matrices (ASM) through the following bijection: A Gog triangle X is mapped to the ASM M such that M_{i,j} = 1 if row i of X contains the value j and this value is not in the previous row; M_{i,j} = -1 if row i of X does not contain the value j and this value is in the previous row.
The set of ASMs corresponding to gapless Gog triangles is invariant by transposition.
LINKS
Ludovic Schwob, Table of n, a(n) for n = 1..31
Ludovic Schwob, Python program
EXAMPLE
The 6 gapless Gog triangles of size 3 and the corresponding ASMs are:
1 2 2 3 2 3
1 2 1 2 1 3 2 3 2 3 1 3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
.
[1 ] [ 1 ] [ 1 ] [ 1] [ 1 ] [ 1]
[ 1 ] [1 ] [1 -1 1] [ 1 ] [ 1] [1 ]
[ 1] [ 1] [ 1 ] [1 ] [1 ] [ 1 ]
Only the first four ASMs are symmetric, giving a(3) = 4.
PROG
(Python) # See Links
CROSSREFS
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Mar 16 2026
STATUS
approved
