A391973 Number of self-dual Catalan triangles of size 2*n+1.

1, 2, 10, 120, 3434, 233876, 37866212

Offset

0,2

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 antiautomorphism given by (X_{i,j}) -> (n + 1 - X_{i,i-j+1}). A Catalan triangle is said to be self-dual if is fixed by this antiautomorphism.

Links

Table of n, a(n) for n=0..6.

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(2) = 10 self-dual Catalan triangles of size 5 are:
      3            3            3            3            3
     2 4          2 4          3 3          2 4          3 3
    1 3 5        2 3 4        2 3 4        2 3 4        2 3 4
   1 2 4 5      1 2 4 5      1 2 4 5      1 3 3 5      1 3 3 5
  1 2 3 4 5    1 2 3 4 5    1 2 3 4 5    1 2 3 4 5    1 2 3 4 5
.
      3            3            3            3            3
     2 4          3 3          2 4          3 3          3 3
    2 3 4        2 3 4        2 3 4        2 3 4        3 3 3
   2 2 4 4      2 2 4 4      2 3 3 4      2 3 3 4      2 3 3 4
  1 2 3 4 5    1 2 3 4 5    1 2 3 4 5    1 2 3 4 5    1 2 3 4 5

Crossrefs

Cf. A390493 (symmetric Catalan triangles), A391969 (Catalan triangles).

Sequence in context: A120597 A322295 A363586 * A256832 A060690 A013038

Adjacent sequences: A391970 A391971 A391972 * A391974 A391975 A391976

Keyword

nonn,more

Author

Ludovic Schwob, Dec 25 2025

Status

approved