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MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatorial Combinatory analysis". At the center was his $\Omega_{\geq}$ operator, for which he proved many properties. He then claimed that plane partitions were a simple toy case to apply these lemmas and was able to compute many interesting generating functions in some limited cases but ran into some problems with the general case of unrestricted plane partitions. He was however lead to some conjectures, some of which he proved later. From there it became clear that there was a lot of interesting mathematics related to plane partitions. I believe you will find some interesting material in the series of papers "MacMahon's partition analysis" I-XII by G.E. Andrews, P.Paule, A Riese and V. Strehl.

Edit: I was a bit rushed to conclude that $\Omega$ had something to do with the motivation to look at plane partitions, see Richard Stanley's answer. I still believe that it was part of the machinery that he built for the same kind of problems that inspired looking at plane partitions. (I mean all of the results about counting tuples of integers satisfying sets of equalities/inequalities.)

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MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatorial analysis". At the center was his $\Omega_{\geq}$ operator, for which he proved many properties. He then claimed that plane partitions were a simple toy case to apply these lemmas and was able to compute many interesting generating functions in some limited cases but ran into some problems with the general case of unrestricted plane partitions. He was however lead to some conjectures, some of which he proved later. From there it became clear that there was a lot of interesting mathematics related to plane partitions. I believe you will find some interesting material in the series of papers "MacMahon's partition analysis" I-XII by G.E. Andrews, P.Paule, A Riese and V. Strehl.

Edit: I was a bit rushed to conclude that $\Omega$ had something to do with the motivation to look at plane partitions, see Richard Stanley's answer. I still believe that it was part of the machinery that he built for the same kind of problems that inspired looking at plane partitions. (I mean all of the results about counting tuples of integers satisfying sets of equalities/inequalities.)

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MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatorial analysis". At the center was his $\Omega_{\geq}$ operator, for which he proved many properties. He then claimed that plane partitions were a simple toy case to apply these lemmas and was able to compute many interesting generating functions in some limited cases but ran into some problems with the general case of unrestricted plane partitions. He was however lead to some conjectures, some of which he proved later. From there it became clear that there was a lot of interesting mathematics related to plane partitions. I believe you will find some interesting material in the series of papers "MacMahon's partition analysis" I-XII by G.E. Andrews, P.Paule, A Riese and V. Strehl.