Degeneracy of vibrational levels

For s equivalent oscillators, the degeneracy of vibrational level v is found from basic probability analysis to equal to the number of unique ways of putting v identical objects into s boxes ... 

Here, we are using the summation index i as the index on the quantum number v. Although a real molecule will fall apart if v, gets too large, we can assume that the summation goes to infinity. We also know that the degeneracies of vibrational levels are 1 as long as they are counted individually—a circumstance we shall meet in polyatomic molecules. 

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