---
title: Durfee Triangle in Ferrers Diagram Partitions
url: https://www.emergentmind.com/topics/durfee-triangle
type: topic
---

# Durfee Triangle in Ferrers Diagram Partitions

Searching arXiv for the cited papers to ground the article in current literature.
arxiv_search.run({"query":"id:2507.20260 OR id:2507.19047 OR id:2206.02778","max_results":10})
In the theory of integer partitions, the **Durfee triangle** is the largest right-angled isosceles triangle contained in the Ferrers diagram of a partition, with the right angle or apex anchored at the top-left corner. For a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$, the Durfee triangle has size $k$ precisely when $\lambda_j\ge k-j+1$ for $1\le j\le k$, equivalently when the diagram contains the staircase subpartition with parts $1,2,\ldots,k$; its area is the triangular number $T_k=1+2+\cdots+k=\frac{k(k+1)}2$ [2507.19047, 2507.20260]. Recent work connects this statistic to rook placements on Ferrers boards, rational generating functions, fixed-length linear recurrences, quasi-polynomial formulas, modular periodicity, and asymptotic laws for partitions with prescribed triangle size [2507.20260, 2507.19047].

## 1. Definition within Ferrers-diagram combinatorics

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_d)$ be a partition with $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_d\ge1$. Its Ferrers or Young diagram consists of $d$ left-aligned rows, with $\lambda_j$ boxes in row $j$ [2507.19047]. In the coordinate convention used in the rook-theoretic treatment, the Ferrers diagram of $\pi=(\lambda_1,\ldots,\lambda_r)$ is the set of nodes at integer coordinates $(i,j)$ with $-r+1\le i\le0$ and $0\le j\le \lambda_{|i|+1}-1$, and the associated Ferrers board $B_\pi$ is obtained by replacing each node with a unit square [2507.20260].

The classical Durfee square is the largest square contained in the Ferrers diagram and has size $k$ when $\lambda_j\ge k$ for $j=1,\dots,k$ [2507.19047]. The Durfee triangle is the corresponding staircase-shaped invariant: it is the largest top-left right-angled isosceles triangle whose horizontal and vertical sides have length $k$ [2507.19047]. Equivalently, a partition has Durfee triangle size $k$ if and only if
$$
\lambda_j\ge k-j+1\qquad (1\le j\le k).
$$
A triangle of size $k$ contains $T_k=\frac{k(k+1)}2$ nodes [2507.19047, 2507.20260].

This definition is strictly different from the Durfee-square condition. The partition $(5,3,2,1)$ of $11$, for example, has Durfee square size $2$ and Durfee triangle size $4$ [2507.20260]. The triangle therefore measures a larger staircase core than the square in many diagrams.

## 2. Rook-theoretic interpretation

A central result of the 2025 literature is that the Durfee triangle admits an exact formulation in terms of non-intersecting rook placements on Ferrers boards. For a Ferrers board $B$, the rook number $R(r,B)$ is the number of ways to place $r$ pairwise non-intersecting rooks on $B$, where non-intersecting means occupying distinct rows and distinct columns; the rook polynomial is
$$
f_B(x):=\sum_{n=1}^{\infty} R(n,B)x^n
$$
[2507.20260].

For a partition $\pi$, the **max-rook number** $R(\pi)$ is defined as the maximal number of non-intersecting rooks that can be placed on $B_\pi$ [2507.20260]. The structural theorem is that $R(\pi)$ equals the size of the Durfee triangle of $\pi$ [2507.20260, 2507.19047]. In the formulation given in the rook-decomposition paper, a Ferrers board admits $k$ non-intersecting rooks if and only if it contains the corresponding $k$-step triangular staircase starting at the top-left corner; this is exactly the condition for a Durfee triangle of size $k$ [2507.20260].

This equivalence reorganizes several counting problems. If $R_k(n)$ denotes the number of partitions of $n$ whose Durfee triangle has size $k$, then the same quantity counts partitions whose Ferrers board has max-rook number $k$ [2507.20260]. Writing $P(n)$ for the partition function, every partition has a unique Durfee triangle, hence
$$
P(n)=\sum_{k=1}^n R_k(n),
$$
a decomposition termed the **rook decomposition** of $P(n)$ [2507.20260].

## 3. Rational generating functions and structural decomposition

For fixed $k$, the generating function
$$
\mathcal F_k(q):=\sum_{n=0}^\infty R_k(n)q^n
$$
is rational [2507.19047]. The main formula is
$$
\mathcal F_k(q)=\frac{q^{T_k}\,\varphi_k(q)}{(q;q)_k},
$$
where $\varphi_k(q)\in\mathbb Z[q]$ is a polynomial of degree $k^2$ with
$$
\varphi_k(q)=1+c_1q+\cdots+c_{k^2-1}q^{k^2-1}+(-1)^{k-1}q^{k^2},
$$
and, when $k$ is odd, $\varphi_k(-1)=0$, so the minimal denominator can be reduced to $(q;q)_k/(1+q)$ [2507.19047]. This is the triangle analogue of the classical Durfee-square identity
$$
\sum_{n=0}^\infty D_k(n)q^n=\frac{q^{k^2}}{(q;q)_k^2},
$$
where $D_k(n)$ counts partitions of $n$ with Durfee square size $k$ [2507.19047].

The rational form arises from a decomposition of the Ferrers diagram at the staircase core. After removing the Durfee triangle of size $k$, one records horizontal excesses $m_1,\ldots,m_d$ in the first $d$ rows and vertical excesses $n_1,\ldots,n_{k-d}$ in the first $k-d$ columns, subject to the constraints
$$
m_1\ge1,\qquad 1\le m_j\le m_{j-1}+1\quad(2\le j\le d),
$$
and
$$
n_1\ge0,\qquad 0\le n_j\le n_{j-1}+1\quad(2\le j\le k-d).
$$
Introducing
$$
A_d(q):=\sum_{k_1=0}^\infty\sum_{k_2=0}^{k_1+1}\cdots\sum_{k_d=0}^{k_{d-1}+1} q^{k_1+\cdots+k_d},\qquad A_0(q)=1,
$$
one obtains the convolution
$$
\mathcal F_k(q)=q^{T_k}\sum_{d=0}^k q^d\,A_d(q)\,A_{k-d}(q).
$$
Moreover,
$$
A_d(q)=\frac{\alpha_d(q)}{(q;q)_d},
$$
where $\alpha_d(q)\in 1+q\mathbb Z[q]$ has degree $(d-1)d$, leading coefficient $(-1)^{d-1}$, and $\alpha_d(1)=1$ [2507.19047]. Clearing denominators with Gaussian binomial coefficients gives
$$
\varphi_k(q)=\sum_{d=0}^k \binom{k}{d}_q q^d\,\alpha_d(q)\,\alpha_{k-d}(q).
$$

The same rationality is visible concretely in the explicit formulas computed for small $k$ in the rook-decomposition paper. For example,
$$
\mathcal F_3(q)=\frac{q^6(1+2q+q^2+2q^3-q^4-q^6-q^7+q^8)}{(1-q)^3(1+q+q^2)},
$$
$$
\mathcal F_4(q)=\frac{q^{10}(1+4q+6q^2+7q^3+6q^4+2q^5-5q^7-5q^8-5q^9+q^{11}+3q^{12}+2q^{13}-q^{16})}{(1-q)(1-q^2)(1-q^3)(1-q^4)},
$$
and an explicit rational form is also given for $\mathcal F_5(q)$ [2507.20260].

## 4. Exact formulas, recurrences, and quasi-polynomial behavior

Because $\mathcal F_k(q)$ is rational, the sequence $(R_k(n))_{n\ge0}$ is C-finite [2507.19047]. More precisely, if $T_k=\frac{k(k+1)}2$, then the recurrence order is $T_k-1$ for odd $k\ge3$ and $T_k$ otherwise [2507.19047]. For $k=3$, the recurrence extracted from the denominator is
$$
R_3(n+5)=2R_3(n+4)-R_3(n+3)+R_3(n+2)-2R_3(n+1)+R_3(n)\qquad(n>9),
$$
equivalently,
$$
R_3(n)=2R_3(n-1)-R_3(n-2)+R_3(n-3)-2R_3(n-4)+R_3(n-5)\qquad(n>14)
$$
[2507.19047, 2507.20260]. For $k=2$, one has
$$
\mathcal F_2(q)=\frac{q^3(1+2q+q^2+q^3-q^4)}{(1-q)^2(1+q)},
$$
and for $n>7$,
$$
R_2(n)=R_2(n-1)+R_2(n-2)-R_2(n-3)
$$
[2507.20260].

For fixed $k$, $R_k(n)$ is eventually a quasi-polynomial of degree $k-1$ and quasi-period $\operatorname{lcm}(1,2,\dots,k)$, valid for all $n>k^2$ [2507.19047]. The case $k=3$ reduces to quasi-period $3$, and for $n>9$ the exact formula is
$$
R_3(n)=
\begin{cases}
6m^2-15m+7,& n=3m,\\
6m^2-11m+2,& n=3m+1,\\
6m^2-7m-1,& n=3m+2.
\end{cases}
$$
An equivalent expression is
$$
R_3(n)=\frac{2}{3}n^2-5n+\frac{59}{9}+\frac{4}{9}\cos\big(2\pi n/3\big)
$$
[2507.19047].

For $k=4$, the rook-decomposition paper gives an exact formula for $n>16$ and $n=12m+a$, $0\le a\le11$:
$$
R_4(n)=
\begin{cases}
192m^3-264m^2+87m-3,& a=0,\\
192m^3-216m^2+44m+5,& a=1,\\
192m^3-168m^2+15m+5,& a=2,\\
192m^3-120m^2-12m+7,& a=3,\\
192m^3-72m^2-25m+4,& a=4,\\
192m^3-24m^2-36m+3,& a=5,\\
192m^3+24m^2-33m-2,& a=6,\\
192m^3+72m^2-28m-3,& a=7,\\
192m^3+120m^2-9m-5,& a=8,\\
192m^3+168m^2+12m-5,& a=9,\\
192m^3+216m^2+47m-3,& a=10,\\
192m^3+264m^2+84m+3,& a=11.
\end{cases}
$$
No exact closed formula for $R_5(n)$ is given there, although the generating function and parity properties are established [2507.20260].

Small values illustrate the onset of these sequences. One has $R_1(1)=1$ and $R_1(n)=2$ for all $n\ge2$; also $R_3(6)=1$, $R_3(7)=4$, $R_3(8)=8$, $R_3(9)=15$, and $R_4(10)=1$, $R_4(11)=5$, $R_4(12)=12$ [2507.19047, 2507.20260].

## 5. Arithmetic and asymptotic properties

The exact formulas for small fixed triangle size yield strong congruence information. For $n>9$ and any $p\ge2$,
$$
R_3(n+3p)\equiv R_3(n)\pmod p,
$$
so the period modulo $p$ is $3p$ [2507.20260]. For $n>16$ and any $p\ge2$,
$$
R_4(n+12p)\equiv R_4(n)\pmod p,
$$
so the period modulo $p$ is $12p$ [2507.20260]. More generally, since $R_k(n)$ is eventually a quasi-polynomial, it is eventually periodic modulo any fixed modulus $M$ [2507.19047].

The parity results obtained from the generating functions are especially explicit. For $n>9$,
$$
R_3(n)\equiv
\begin{cases}
0 \pmod 2,& n\ \text{odd},\\
1 \pmod 2,& n\ \text{even},
\end{cases}
$$
while for $n>16$,
$$
R_4(n)\equiv
\begin{cases}
0 \pmod 2,& n\equiv 4,6 \pmod 8,\\
1 \pmod 2,& n\equiv 0,1,2,3,5,7 \pmod 8,
\end{cases}
$$
and for $n>25$,
$$
R_5(n)\equiv
\begin{cases}
0 \pmod 2,& n\equiv 5,7 \pmod 8,\\
1 \pmod 2,& n\equiv 0,1,2,3,4,6 \pmod 8.
\end{cases}
$$
The same source remarks that for $n>16$ and $n$ odd, $R_4(n)$ is always odd, and for $n>25$ and $n$ even, $R_5(n)$ is always odd [2507.20260].

Asymptotically, fixed triangle size leads to polynomial growth. The general theorem is
$$
R_k(n)=\frac{2^k}{k!(k-1)!}\,n^{k-1}+O(n^{k-2}),
\qquad
R_k(n)\sim \frac{2^k}{k!(k-1)!}\,n^{k-1},
$$
as $n\to\infty$ for fixed $k$ [2507.19047]. The previously derived cases agree with this formula:
$$
R_3(n)\sim \frac23 n^2,\qquad R_4(n)\sim \frac19 n^3
$$
[2507.20260]. In comparison, for fixed $k$, the Durfee-square count $D_k(n)$ has degree $2k-1$ rather than $k-1$ [2507.19047].

## 6. Relation to Durfee squares, $k$-measures, and terminological boundaries

The Durfee triangle sits naturally beside the Durfee square but is not interchangeable with it. For Durfee squares, the generating function is especially simple:
$$
\sum_{n=0}^\infty D_k(n)q^n=\frac{q^{k^2}}{(q;q)_k^2},
$$
whereas for Durfee triangles the denominator is a single $(q;q)_k$ and the numerator is a nontrivial polynomial $\varphi_k(q)$ of degree $k^2$ with $\varphi_k(1)=2^k$ [2507.19047]. The square count therefore arises from two independent bounded partition structures, while the triangle count is governed by coupled “step at most $+1$” excess conditions after staircase removal [2507.19047]. This distinction explains why fixed-size Durfee triangles and fixed-size Durfee squares have different growth degrees and different recurrence orders.

A separate line of work on $k$-measures of partitions provides an important contrast. The $k$-measure of a partition is the length of the largest subsequence of parts whose consecutive differences are at least $k$ [2206.02778]. For $k=2$, the number of partitions of $n$ with $2$-measure $m$ equals the number with Durfee square of side $m$, and the paper proves this bijectively [2206.02778]. For general $k\ge3$, however, the natural shape in that framework is not a triangle but a **$(k,m)$-Durfee polygon**: if $k$ is even or $m$ is odd, this is an $m\times\bigl(1+\frac{k(m-1)}2\bigr)$ rectangle; if $k$ is odd and $m$ is even, it is a two-step rectangle with $m/2$ rows of width $\frac{k(m-1)+3}{2}$ and $m/2$ rows of width $\frac{k(m-1)+1}{2}$ [2206.02778]. In particular, for $k=3$ that paper explicitly states that the associated shape is rectangular when $m$ is odd and two-step rectangular when $m$ is even, not triangular [2206.02778].

This distinction addresses a common misconception. The term **Durfee triangle** belongs to the 2025 staircase-based theory of partitions and Ferrers-board rook maxima [2507.20260, 2507.19047]. It is not the terminology used in the $k$-measure literature, where the correct generalization is the $(k,m)$-Durfee polygon [2206.02778]. The two frameworks are related by their shared concern with top-left anchored substructures in Ferrers diagrams, but they are not equivalent shape theories.

Source: https://www.emergentmind.com/topics/durfee-triangle