As I have previously discussed with colleagues, it can be difficult to find information on numbers of partnerships and other UK businesses. The main source is the Business Population Estimates, which is published annually - latest edition at https://www.gov.uk/government/statistics/business-population-estimates-2018 - and information on methodology at https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/746602/OFFICIAL_SENSITIVE_-_BPE_2018_-_methodology_note_FINAL_FINAL.pdf
The House of Commons Library has now published a useful statistical analysis at: http://researchbriefings.files.parliament.uk/documents/SN06152/SN06152.pdf