Abstract
In his recent work Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors.
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Notes
This way of drawing Ferrers graphs is usually called the “English notation.”
This parameter j appeared implicitly in the bijection of Alladi and Gordon; see [25, p. 26, Sect. 4.4].
Strictly speaking, the images under the mapping \(\phi \) should be words ending with x or z. So, when \(\hat{u}\) ends with y, we simply ignore all its ending y’s (doing this will not change the weight of \(\hat{u}\)), and then apply \(\phi ^{-1}\).
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The author is grateful to the anonymous referee for several helpful suggestions.
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The author was supported by the National Natural Science Foundation of China Grant 12171059 and the Natural Science Foundation Project of Chongqing (No. cstc2021jcyj-msxmX0693)
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Fu, S. Combinatorial proofs and refinements of three partition theorems of Andrews. Ramanujan J 60, 847–860 (2023). https://doi.org/10.1007/s11139-022-00607-y
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DOI: https://doi.org/10.1007/s11139-022-00607-y