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bibkey: meeussen2004a098052 authors: Wouter Meeussen year: 2004 title: “OEIS A098052, T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways; with its twin A098530” doi: null url: https://oeis.org/A098052 claim: “The entries tabulate solid partitions of n by their number of extensions to n+1 (A098052) and solid partitions of n+1 by their number of shrinkings to n (A098530), and conjecture that the first column of each is A007426 = tau_4.” strata_touched:

  • D5/S3/Combinatorics/SolidPartitionFirstColumn license: citation-only triage: anchor

OEIS A098052 and A098530

A098052 (Wouter Meeussen, 2004-09-11) is the triangle

T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

with the COMMENTS lines

Row sums are A000293 (solid partitions) by definition.

First column is conjectured to be A007426 = tau_4(n).

All solid partitions can be extended in at least 4 ways (hence the offset 4).

Its table begins 1; 4; 4,6; 10,12,0,4; 4,30,12,12,0,0,1; 16,48,18,48,0,6,4, so the first column (k = 1, exactly four extensions) is 1, 4, 4, 10, 4, 16 for n = 1, …, 6.

A098530 (Wouter Meeussen, 2004-09-12) is the triangle

T(n,k) counts solid partitions of n+1 that can be ‘shrunk’ in k ways to a solid partition of n by removing 1 element from it. Equivalently, it counts how many solid partitions of n+1 have k different solid partitions of n it just covers.

with the COMMENTS line

Sequence starts 4; 4,6; 10,12,4; 4,42,12,1; 16,60,60,4; 4,105,164,34; Row sums are A000293= the solid partitions of n+1 apart from offset. First column conjectured to be the (beheaded) A007426.

A007426 is

d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.

Verified locator

  • URL: https://oeis.org/A098052 (last modified 2025-02-03, COMMENTS) and https://oeis.org/A098530 (last modified 2012-03-30, COMMENTS); both retrieved 2026-09-27.
  • Related entries: https://oeis.org/A007426, https://oeis.org/A000293.