The complaint does not present evidence that anyone who lives in the school zone has been prevented from enrolling, but it portrays Mr. Dinowitz’s focus on enrollment as tinged with bias.  It cites a meeting in November 2009 between Mr. Dinowitz, Mr. Verdi and Dr. Connelly, the former principal, who had recently arrived at the school. According to the complaint, Mr. Dinowitz said that people were misrepresenting their addresses to get into P.S. 24 and that he could tell which children were not from Riverdale “by the way they walk, talk and wear their pants.”  In an interview on Tuesday, Mr. Dinowitz said he “never said anything like that,” and he characterized Mr. Verdi’s complaint as “lie after lie.”  “This is about overcrowding, period,” he said. “That’s it. That’s what this is all about.”  Dr. Connelly, however, said that she recalled Mr. Dinowitz using those words.  “He was referring to kids who were not from the community and kids that were black or Hispanic,” she said.  P.S. 24 enrolls children from kindergarten through fifth grade. The school’s population is 42 percent white, 41 percent Hispanic, 8 percent Asian and 7 percent black. Twenty-seven percent of students receive free or reduced-price lunch. On the most recent round of state tests, 57 percent of third through fifth graders were proficient in reading, while 62 percent were proficient in math, compared with 38 percent and 36 percent citywide.  The elementary schools in nearby Kingsbridge are all largely Hispanic, and the vast percentage of their students receive free or reduced-price lunch. One, P.S. 207, which serves prekindergarten through second grade, is on the state’s list of persistently dangerous schools. While one of the schools, the Milton Fein School, also does well on annual state tests, the other school with children of test-taking age lags.  P.S. 24’s enrollment has increased dramatically in the past decade, rising to 1,030 students this year from 715 students in 2006-7. The department said the building was now at 122 percent capacity.  Dr. Connelly said that the increase, which began before she arrived, was caused by population growth, not a rise in the number of non-zoned students being enrolled.  According to the Education Department, the percentage of non-zoned children enrolled in kindergarten at P.S. 24 fell to about 10 percent in 2015-16 from 13 percent in 2010-11 (not counting the gifted and talented program), while the number of zoned students in kindergarten has grown by 36 percent.  Dr. Connelly said that some non-zoned students were assigned to P.S. 24 by the department. In other cases a parent might come into the school around November, hoping to transfer a child from a low-performing school. If there were seats available in the student’s grade, Dr. Connelly said, she would sometimes let the student in.  She said that accounted for only a small number of children.  “But it was still looked upon as some kind of a threat that I was letting children into the school that don’t belong at P.S. 24,” she said.  Mr. Dinowitz disagreed that population growth could account for the increase in enrollment.  “There has not been a population boom in the neighborhood,” he said.  Mr. Dinowitz’s own two children attended P.S. 24 in the 1990s, even though he lived outside the school zone.  Asked about that, he said it was irrelevant to the present situation, because the school was underused at the time.  “If the school had empty seats, fine, but the school doesn’t have empty seats,” he said.  In the 2015-16 school year, the debate over P.S. 24’s enrollment increase reached a crisis when the school lost its lease on an annex that housed fifth graders. Amid finger-pointing and a dispute about whether Dr. Connelly threw out teachers’ desks, she abruptly announced that she was retiring in October last year.  After her departure, Mr. Dinowitz and others, including the district superintendent, Ms. Mashel, and Ms. Feldman, the interim principal, met at the school in January this year, according to Mr. Verdi’s complaint and a report from the Education Department’s Office of Special Investigations. Mr. Dinowitz again complained about out-of-zone students.  It was then suggested that someone from Mr. Dinowitz’s office could go to the school during the registration process in late March and early April.  Randi Martos, Mr. Dinowitz’s chief of staff, ended up taking part and reviewed families’ documents. Department policy requires parents to provide two documents showing proof of residence, but parents were told they needed three, according to Mr. Verdi’s lawsuit and the Education Department’s investigative report. There is no evidence any zoned families were turned away.  The department’s investigation determined that Ms. Mashel had shown poor judgment and that Ms. Feldman had failed to supervise the registration process and allowed Ms. Martos to review parents’ and students’ personal information. Neither woman responded to a phone call requesting comment.  Despite his concerns about overcrowding, Mr. Dinowitz has opposed calls to build an addition to the school. In the interview, he said the overcrowding problem could be solved simply by enforcing enrollment policies.  The battles have left parents frustrated. Bob Heisler, a former president of the parent association, said he did not believe Mr. Verdi’s charges of racial bias, but he criticized Mr. Dinowitz and other local politicians for showing “no leadership over the years” in dealing with the increased demand for school seats.  “I personally feel that some of the local politicians want to keep Riverdale as they have seen it — the way they remember it from their own childhood — and they’re not open to the changes in the demographics that are going on in that community,” he said.  A version of this article appears in print on October 6, 2016, on page A25 of the New York edition with the headline: Bias Allegation Looms Over Bronx School as Administrators Exit. Order Reprints| Today's Paper|Subscribe
An assistant principal has filed a complaint against the city’s Education Department saying that a local assemblyman’s involvement with a school’s enrollment was rooted in bias.