A central limit theorem for integer partitions into small powers

Abstract

The study of the well-known partition function p(n) counting the number of solutions to n=a1+⋯+aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = a_{1} + \\dots + a_{\\ell }$$\\end{document} with integers 1≤a1≤⋯≤aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le a_{1} \\le \\dots \\le a_{\\ell }$$\\end{document} has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=a1α+⋯+aℓα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} n=\\left\\lfloor a_1^\\alpha \\right\\rfloor +\\cdots +\\left\\lfloor a_\\ell ^\\alpha \\right\\rfloor \\end{aligned}$$\\end{document}with 1≤a1<⋯<aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le a_1< \\cdots < a_\\ell $$\\end{document} and some fixed 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\alpha < 1$$\\end{document}. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.