A390493 Number of symmetric Catalan triangles of size n.

1, 1, 2, 4, 12, 36, 168, 768, 5556, 38904, 435084, 4676496, 80726904, 1333343808, 35496821520, 901465150464, 36993462332688, 1445037903407448, 91378480764800484, 5491575111929758560, 535003999187581696752, 49474325864424836136672, 7424560895933543838815304, 1056608189903956740657033312

Offset

0,3

Comments

A Catalan triangle of size n is a Gelfand-Tsetlin pattern with bottom row 1 2 ... n which is gapless on diagonals and antidiagonals (entries increase by at most 1 on diagonals and antidiagonals). The lattice of Catalan triangles of size n has an automorphism given by (X_{i,j}) -> (X_{i,i-j+1} - i + 2*j - 1). A Catalan triangle is said to be symmetric if is fixed by this automorphism.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..30

Florent Hivert, Vincent Pilaud, and Ludovic Schwob, Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra, arXiv:2511.06968 [math.CO], 2025. See Table 2 p. 25.

Example

The a(4) = 12 symmetric Catalan triangles of size 4 are:
     1          2          2          3          2          3
    1 2        1 2        2 3        2 3        2 3        2 3
   1 2 3      1 2 3      1 2 3      1 2 3      1 3 3      1 3 3
  1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4
.
     2          3          2          3          3          4
    2 3        2 3        2 3        2 3        3 4        3 4
   2 2 4      2 2 4      2 3 4      2 3 4      2 3 4      2 3 4
  1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4    1 2 3 4

Crossrefs

Cf. A391969 (Catalan triangles), A391973 (self-dual Catalan triangles).

Sequence in context: A003701 A255432 A275539 * A356062 A193049 A114500

Adjacent sequences: A390490 A390491 A390492 * A390494 A390495 A390496

Keyword

nonn

Author

Ludovic Schwob, Dec 25 2025

Status

approved