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- W82092628 abstract "For positive integers k,a,n and an even integer λ, let Aλ,k,a(n) denote the number of partitions of n into parts such that no part that is not congruent to 0 (mod λ+1) may be repeated and no part is congruent to 0 or ±(λ+1)(a−λ/2) (mod (2k−λ+1)(λ+1)). For an odd integer λ, let Aλ,k,a(n) denote the number of partitions of n into parts such that no part that is not congruent to 0 (mod (λ+1)/2) may be repeated, and such that no part is congruent to λ+1 (mod 2λ+2), and such that no part is congruent to 0, or ±(2a−λ)(λ+1)/2 (mod (2k−λ+1)(λ+1)). Let Bλ,k,n(n) denote the number of partitions of n of the form b1+⋯+bs with bi≥bi+1, no part that is not congruent to 0 (mod λ+1) is repeated, bi−bi+k−1≥λ+1 with strict inequality if λ+1∣bi and fj+⋯+fλ−j+1≤a−j for 1≤j≤(λ+1)/2 and f1+⋯+fλ+1≤a−1, where fi is the number of appearances of i in the partition. In 1974, G. E. Andrews conjectured in [On the general Rogers-Ramanujan theorem, Amer. Math. Soc., Providence, R.I., 1974] and proved in [G. E. Andrews, C. Bessenrodt and J. B. Olsson, Trans. Amer. Math. Soc. 344 (1994), no. 2, 597–615] that A4,3,3(n)=B04,3,3(n), where B04,3,3(n) is the number of partitions enumerated by B4,3,3(n) with the added conditions f5j+2+f5j+3≤1 for j≥0, f5j+4+f5j+6≤1 for j≥0, and f5j−1+f5j+f5j+5+f5j+6≤3 for j≥1. The authors of the paper under review prove the next natural case when k≤λ: For all n≥0, A5,3,3(n)=B05,3,3(n), where B05,3,3(n) is the number of partitions enumerated by B5,3,3(n) with the following conditions: no part is congruent to 3 (mod 6), f6j+2+f6j+4≤1 for j≥0, f6j+5+f6j+7≤1 for j≥0, and f6j−1+f6j+f6j+6+f6j+7≤3 for j≥1." @default.
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- W82092628 date "2005-01-01" @default.
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- W82092628 title "A new theorem on partitions" @default.
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