What is the difference between ekadandi ("Single staff”) and tridandi ("triple staff") Sannyasis?
Are Tridandis Vaishnavites?
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A commentary on a Bhagavata Purana verse clarifies that Vaishnava Sannyasis are Tridandis.
The Vaishnava sannyasi is known as a tridandi-sannyasi. The Mayavadi sannyasi accepts only one danda, not understanding the purpose of tri-danda
Scriptures like Daksha Smriti also talk about triple staff holder Sannyasis.
Many persons drive their livelihood Under the Umbrage of a triple start [i.e., of being a Sannyasin]; he, who does not know Brahman is not worthy of holding the triple staff.
Chapter 7; Verse 30
However, as shown in this answer, various so-called Sannyasa Upanishads prescribe differently depending on the type of Sannyasi one is. For example, the type who has achieved the highest state of spiritual evolution is not required to carry a staff. Hamsas, who are also highly evolved, are required to carry only one Danda.
Bahudakas are those who carry a triple staff, a water pot, a sling , a shoulder yoke, a water strainer, a bowl, shoes, and a seat; who wear a topknot, a sacrificial string, a loincloth, and an ochre garment; who beg their food from virtuous Brahmin households; and who thus seek the self
Hamsas are those who carry a single staff; who wear a sacrificial string but not a topknot; who carry in their hand a sling and a water pot; who spend only one night in a village and five nights in a town or a sacred bathing place; who perform penances such as the one- and two-day fasts, the krcchra, and the lunar fast; and who thus seek the self.
Paramahamsas are those who carry no staff; who are shaven-headed; who wear a loincloth and a patched garment; whose emblem and conduct are concealed
The Narada Parivrajaka Upanishad (linked to the Atharva Veda), in it's Chapter 5, however emphasises that just by carrying a single staff one does not attain enlightenment. The real staff that the Sannyasi needs to carry is the "staff of wisdom". One who truly carries this staff of wisdom is called the Ekadandi.
V-27. If an ascetic merely lives on the alms devoid of the qualities of forbearance, wisdom, dispassion, tranquillity, etc., he is a bane of the conduct of ascetics. V-28. Not by bearing an (emblematic) staff, not by a shaven head, not by (special) dress, not by hypocritical airs (of sage-hood) does liberation (come to one).
V-29. He who bears the staff of wisdom is said to be ‘single staffed’. The ascetic who carries a wooden stick, eats all sorts of food and is devoid of wisdom goes to terrible hells called Maharaurava. V-30. A stable position (in a monastery) is said by great sages to be similar to the excreta of a sow; hence, leaving it aside, the ascetic shall move about like an (assiduous) worm. V-31. The Turiyatita ascetic shall have food and clothing without solicitation and just as they happen to occur, by others will. He shall be unclad and have a bath at others’ will. V-32. The ascetic whose behaviour is well in harmony even with the dreaming state as with the waking state, is considered the best; he is the most excellent among those that follow the Vedanta. V-33. In non-acquirement (of alms) he shall not grieve; in its acquisition he shall not feel joy. Avoiding attachment to material things he shall simply keep himself alive (for a higher purpose).