A094866 Number of truncated ST-pairs O(q^n).

1, 2, 4, 6, 11, 15, 26, 41, 67, 96, 138, 197, 300, 431, 636, 893, 1258, 1723, 2447, 3425, 4962, 6839, 10000, 13989, 21383, 30781, 48292, 70456, 110214, 159686, 253265, 374385, 591648, 876405, 1354888

Offset: 3

Author

Barry Cipra, Jun 15 2004

Keywords

• nonn

Comments

A truncated ST-pair O(q^n) consists of a subset S of {1, 2, ..., n-1} and a subset T of {1, 2, ..., n-2} such that (Product_{k in S} 1/(1-q^k)) - q (Product_{k in T} 1/(1-q^k)) = 1 + O(q^n). - Andrey Zabolotskiy, Feb 27 2024

References

• F. G. Garvan, Shifted and Shiftless Partition Identities, in Number Theory for the Millennium II (M. A. Bennett et al., eds.), AK Peters, Ltd. 2002, pp. 75-92.

Links

• F. G. Garvan, Shifted and shiftless partition identities (2001).

Programs

Mathematica

st[n_] := Select[Flatten[Table[{s, t}, {s, Subsets@Range[n - 1]}, {t, Subsets@Range[n - 2]}], 1], Normal[Product[1/(1-q^k) + O[q]^n, {k, First@#}] - q Product[1/(1-q^k) + O[q]^n, {k, Last@#}] - 1] == 0 &];
Table[Length@st[n], {n, 3, 9}] (* Andrey Zabolotskiy, Feb 27 2024 *)