A115965 Number of planar subpartitions of the size n pyramidal planar partition.

1, 2, 9, 96, 2498, 161422, 26217833, 10794429504, 11337432232915, 30523827764041406, 211454683487501523010, 3780653093135870722425698, 174888389542335169590721957741, 20974304532067473149987767753711780, 6532663526470788562984929108703145557489

Offset

0,2

Comments

This is a 2-dimensional analog of the Catalan numbers C_n (A000108). The number of subpartitions of the triangular partition [n,n-1,...,1] is C_{n+1}. The planar partition having its subpartitions counted is:

n n-1 ... 2 1

n-1 n-2 ... 1

... ...

2 1

1

Links

Ludovic Schwob, Table of n, a(n) for n = 0..18

Oleg Lazarev, Matt Mizuhara, and Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions

Ludovic Schwob, Python Program

Example

The 9 planar subpartitions of [2,1|1] are [], [1], [2], [1,1], [1|1], [2,1], [2|1], [1,1|1] and [2,1|1] itself, so a(2)=9. (Here "," separates values on the same line and "|" separates lines.)

Prog

(Python) # See Links - Ludovic Schwob, Jan 24 2026

Crossrefs

Cf. A115728, A115729, A000219, A000108, A008793.

Sequence in context: A354316 A345466 A381376 * A001142 A111847 A013132

Adjacent sequences: A115962 A115963 A115964 * A115966 A115967 A115968

Keyword

nonn

Author

Franklin T. Adams-Watters, Mar 14 2006

Extensions

Terms a(8) and beyond from Ludovic Schwob, Jan 24 2026

Status

approved