This is the 17866th most frequent Polish word.
łańcuta
"Łańcuta" does not have a specific meaning in Polish; it might be a misspelling or refer to "Łańcut," a town in southeastern Poland.
'Łańcucie' in the sentence refers to the locative form of the proper noun 'Łańcut', the name of a town in Poland, used to indicate the location of the castle.
Zamek w Łańcucie jest jednym z najpiękniejszych w Polsce.
The castle in Łańcut is one of the most beautiful in Poland.
'Łańcucie' again serves as the locative case of 'Łańcut', denoting the location of the event, a ball.
Na balu w Łańcucie bawiono się do białego rana.
At the ball in Łańcut, people enjoyed themselves until dawn.
'Łańcuta' here represents the genitive form of 'Łańcut' because of its grammatical relationship with the word 'ulicami' (the streets of Łańcut).