What is the number of years of Zakat due on the estate of a father who passed away in Safar 1430 AH, and had not paid Zakat since Jumada al-Akhirah 1424 AH, and how is Zakat calculated for these years, and is it calculated on a diminishing basis?

Your father fell short by not paying Zakat. You should strive to seek forgiveness for him and to pay the Zakat for all the years he missed, for it is a debt upon him that cannot be absolved except by its payment, as the Prophet, peace and blessings be upon him, said: "The debt owed to Allah is more worthy of being fulfilled." You should pay the Zakat for five years.

Regarding how to pay it and whether the amount paid for each year should be deducted from the subsequent year: This depends on the juristic disagreement over whether Zakat is due on the asset itself or on the person's liability (dhimmah). If we say it is due on the asset, then you deduct what is paid for the first year from the second, and so on. If we say it is due on the liability, then you pay Zakat for all the years on the entire wealth, and this is more cautious. The esteemed scholar Al-Uthaymeen favored the view that it is due on the asset itself and also relates to the liability (dhimmah), meaning that it is due on the wealth and the person is obligated by it in their liability.