A 76-year-old man died on Friday after being shoved down the stairs at the 18th Street subway station in Manhattan, and the police arrested a suspect who had been arrested multiple times in recent months and had been discharged from Bellevue Hospital’s psychiatric ward just hours before.

The victim, Ross Falzone, landed on his head at the bottom of the stairs and suffered a traumatic brain injury, a fractured spine and a fractured rib after a stranger rushed forward and pushed him, the police said.

Mr. Falzone had been walking north on Seventh Avenue toward the subway station in the Chelsea neighborhood on Thursday evening, said Brad Weekes, assistant commissioner of public information for the Police Department. Walking about 30 yards behind him was the stranger, according to surveillance footage from the scene, Mr. Weekes said. As Mr. Falzone reached the station, the man rushed forward and pushed him down the stairs. He was taken to Bellevue where he died shortly before 3 a.m. on Friday.

The death sparked outrage at City Hall. Mayor Zohran Mamdani quickly called for an investigation into how Bellevue handled the discharge of the suspect and suggested that institutional problems at the hospital might have led to the random attack.

“I am horrified by the killing of Ross Falzone and the circumstances that led to it,” Mr. Mamdani said in a news release on Friday, in which he ordered “an immediate investigation on what steps should have been taken to prevent this tragedy.”

Police identified the suspect as Rhamell Burke, 32.

In the three months preceding the attack, Mr. Burke was arrested four times, Mr. Weekes said, including an arrest on Feb. 2 in connection with an assault on a Port Authority police officer.

Mr. Burke’s most recent interaction with the police began at around 3:30 p.m. Thursday, when he approached a group of N.Y.P.D. officers outside the 17th Precinct station house on East 51st Street, Mr. Weekes said. He grabbed a stick from a pile of garbage on the street and approached the officers, who told him to drop the stick. When he did, officers placed Mr. Burke in a police vehicle and drove him to Bellevue, where he was admitted to the emergency room at around 3:40 p.m., Mr. Weekes said. Mr. Burke was taken to the hospital’s Comprehensive Psychiatric Emergency Program for evaluation and treatment, Mr. Weekes said, and was released from the hospital one hour later.

He was just a mile and a half from the hospital when he encountered Mr. Falzone at around 9:30 p.m. Thursday.

On Friday afternoon, police officers found Mr. Burke in Penn Station, where they arrested him. He was in custody on Friday evening. It was unclear Friday if Mr. Burke had a lawyer.

The mayor said he had requested help from the New York State Department of Health, which will investigate the decision to release Mr. Burke from Bellevue and conduct a review of similar cases at the hospital. The state agency also will investigate psychiatric evaluation and discharge procedures across NYC Health and Hospitals, the city’s public hospital system, according to the news release.

Mr. Falzone was a retired high school teacher who lived alone for many years in an apartment building on the Upper West Side. His friends were in shock on Friday about his death. They shared memories of an affable but private man who rarely spoke about his family or personal life.

Mr. Falzone had been recovering from a recent surgery and seemed more mobile and happy, said Marc Stager, 78, Mr. Falzone’s next-door neighbor on a tree-lined block of West 85th Street. He was known as a cheerful “yapper,” said Briel Waxman, a neighbor. He was the kind of New Yorker who enjoyed chatting with neighbors about historical details of his building and seeing performances at Lincoln Center with friends.

“He was always out and about,” said Ms. Waxman, 35, who often returned to her apartment at midnight or 1 a.m. to find Mr. Falzone arriving home at the same time. “I was like, ‘I don’t know if I’m proud of you or embarrassed of myself,’” she remembered telling him.

Mr. Falzone had wide taste in music — opera, classical, jazz, pop — and neighbors could tell he was home when they heard notes escaping from under his apartment door, Mr. Stager said.

He was “a helpless old guy,” said Mr. Stager, who added that he was “disappointed and shocked, frankly, that somebody could do such a thing” as shove such a defenseless person down the stairs.

When Ms. Waxman moved into the building five years ago, Mr. Falzone was among the first people to welcome her, she said. He once brought a package to her door that had been delivered to the wrong unit and shared that what is now a blank wall in her apartment had once been a fireplace.

Ms. Waxman sat in her living room on Friday and cried as she talked, dabbing her eyes with a tissue. She remembered Mr. Falzone as “just overall, nice, talkative, genuine human.”

A suicide note purported to be written by the sexual predator Jeffrey Epstein while he was in jail in 2019 uses language that in some cases echoes his past writings to friends and family.

One phrase found in the apparent suicide note — “No Fun” — also appears on a handwritten page found in Mr. Epstein’s jail cell at the time of his death, as well as in emails he sent over the years.

And another saying in the suicide note — “watcha want me to do — bust out cryin!!” — appears in emails that Mr. Epstein had written to people close to him.

A cellmate claimed that Mr. Epstein left the suicide note before he was found unresponsive in their cell weeks before his death. The New York Times reported on the note last week and successfully asked a federal judge to unseal it.

If authentic, the note gives a view into Mr. Epstein’s mind-set before he was found dead at age 66 in August 2019. The New York City medical examiner ruled his death a suicide.

‘NO FUN’

A different handwritten note was found in Mr. Epstein’s cell when he died, and investigators believed it was written by him. In that document, Mr. Epstein complained about jail conditions — burned food, giant bugs and being kept in a locked shower. He concluded it with the underlined phrase, “NO FUN!!”

Mr. Epstein also used the phrase in emails when describing things he was unhappy about, or situations that had not gone his way.

‘watcha want me to do — bust out cryin’

Mr. Epstein used the phrase “watcha want me to do — bust out cryin” with friends, and in messages to his brother, Mark Epstein.

Like the note released by the judge, Mr. Epstein’s emails were often short, with staccato phrases and erratic punctuation. The emails were contained in millions of pages of documents the Justice Department released in response to a law passed last year requiring disclosure of records pertaining to Mr. Epstein.

It has become an article of faith in the New York State Capitol that when Gov. Kathy Hochul enters the Red Room on the building’s second floor to announce a budget agreement, the deal is actually far from sealed.

This year was no different.

Despite declaring that “today is the day” to announce an agreement on a $268 billion state budget, Ms. Hochul on Thursday acknowledged that several key initiatives — including a new tax surcharge on multimillion-dollar second homes in New York City — had been agreed on in principle, but that the details still needed work.

Even the top-line figure had not been finalized.

Lawmakers are fond of saying that the devil is in the details. But in the absence of the lengthy budget bills that include those details, which have yet to be printed and voted on, a host of unanswered questions remain.

Here are five of them:

Why did Hochul announce a deal when one hadn’t really been made yet?

New York’s opaque budget process, which starts in January with the State of the State address and is supposed to be completed by April 1, has become far more than a negotiation over a fiscal document.

Governors have tended to use the budget to wedge in legislative priorities, wielding their leverage over billions of dollars to get their way.

Ms. Hochul has embraced this practice. And, in a re-election year, she wanted to convey to voters that she intended to stand up to President Trump’s immigration crackdown, help out New York City and lower costs for everyday New Yorkers.

She made that case on Thursday at a news conference flanked by several of her top aides. Notably missing were the leaders of the State Assembly and Senate.

When will the budget actually be passed?

Not this week. The Assembly speaker, Carl E. Heastie, said on Thursday that it was “very premature” of the governor to say a deal had been reached. He would not even say that the Legislature had agreed to the $268 billion figure.

He complained about Ms. Hochul’s penchant for jamming nonfiscal policies into the budget and said he would not discuss such matters with his members until he had a better sense of the total amount the state would be spending.

As he spoke, members of the Senate and Assembly, who are currently not being paid, were wrapping up their legislative business for the week in a rush to return to their districts. They will be back in Albany on Monday; it is unclear what bill language, if any, will have been printed and distributed by then.

Did Zohran Mamdani get what he wanted?

Mr. Mamdani, the mayor of New York City, campaigned on wresting more than $10 billion in tax increases from the state to pay for his ambitious agenda. That will not happen this year.

Ms. Hochul did accede to a new tax on second homes that targets the city’s richest property owners whose primary residences are outside New York City. The goal is to raise $500 million each year, which will go toward closing the city’s estimated $5.4 billion budget deficit.

But she spurned the mayor’s request to make changes to a tax credit called the Pass Through Entity Tax that is used by some business owners. Mr. Mamdani had said that the measure, which was also backed by the City Council speaker, Julie Menin, could raise up to $1 billion a year in tax revenue.

Aside from tax increases, Mr. Mamdani’s overarching priority has been expanding child care in the city. Ms. Hochul’s budget does just that, with $4.5 billion allotted for child care and prekindergarten programs across the state.

It’s not the whole loaf, or even half. But Mr. Mamdani can point to that funding and say that he is advancing toward his goal of providing free child care for every New York City child under 5. And while the governor rejected his efforts to fund a program to make buses free, she directed more than $1 billion in additional aid to the city that, combined with revenue from the second-home tax and other proposed measures like delays in pension payments, could help Mr. Mamdani work to close its budget gap.

How will the tax on pieds-à-terre work?

State lawmakers — and just about everyone else — are scratching their heads about the details of this tax surcharge, which Ms. Hochul proposed with great fanfare last month. The New York Times previously reported that one proposal being discussed would apply one tax rate to pieds-à-terre with values between $5 million and $15 million; a higher rate for ones valued between $15 million and $25 million; and an even higher rate for properties valued at $25 million or more, according to three people familiar with the matter.

How much the property owners would pay is still up in the air. Ms. Hochul said on Thursday that more details would be coming in the near future and that the tax would apply to units worth $5 million or more.

Also being sorted out is how, exactly, the value of each co-op or apartment would be determined.

“It’s going to take some time to get to the right number to assess that,” the governor said, noting the city’s complex system for calculating a property’s assessed value.

“We’re looking at the difference between what is currently assessed but what is market value,” she added. “We’re working it out with the city. We have had some really good conversations.”

How will pensions change for state workers?

Facing pressure from the state’s largest public unions, Ms. Hochul has been trying to determine how to restore certain pension benefits that had been cut for public employees hired after 2012.

Any changes could end up costing the state hundreds of millions of dollars, while also saddling local municipalities and school districts with increased spending burdens. Several of the labor groups have prioritized lowering the minimum retirement age to 55 from 63.

Ms. Hochul said on Thursday that the particulars were still being negotiated, but stressed that the cost to the state and local governments would be less than the $1.5 billion that has been requested by the unions.

“We are willing to look at this and make changes, but a much more scaled-back monetary proposal,” she said.

“We will release these numbers as soon as it’s absolutely done,” she added.