This number is equivalent to the number of plans (i.e. structural solutions) of the open shop problem with n jobs and n machines - see problems in scheduling theory.

REFERENCES

H. Braesel, M. Harborth, T. Tautenhahn and P. Willenius, On the set of solutions of the open shop problem, Ann. Oper. Res. 92 (1999) 241-263

M. Harborth, Structural analysis of shop scheduling problems, PhD thesis, Otto-von-Guericke-Univ. Magdeburg, GCA-Verlag, 1999 (in German)