Triangular Sum Over Partitions

• Triangular sums over partitions are nested summations with strict index inequalities that capture combinatorial structures and partition statistics.
• They reduce complex nested expressions to explicit finite sums, unveiling key partition identities, generating functions, and connections to triangular numbers.
• These methods offer powerful insights for algebraic combinatorics, number theory, and spectral minimal partition problems through clear computational frameworks.

A triangular sum over partitions refers to any summation structure where indices are nested with strict inequalities, often reflecting the combinatorics of integer partitions and closely related to triangular array regions, figurate numbers, and associated algebraic and analytic objects. Such sums appear throughout algebraic combinatorics, partition theory, and mathematical physics, connecting partition identities, multiple zeta values, and spectral minimal partition problems. The term encompasses both the formal structure of nested sums and results concerning representations or properties of numbers and functions expressible via sums over partitions constrained by triangular or figurate number patterns.

1. Triangular (Nested) Sums: Definition and Partition Indexing

A prototypical triangular sum is the multiple (nested) sum

$$P_{m,q,n}(a_{(1)},\dots,a_{(m)}) = \sum_{q \le N_1 < N_2 < \cdots < N_m \le n} a_{(1);N_1}\, a_{(2);N_2} \cdots a_{(m);N_m},$$

where $\{a_{(i);N}\}_{N}$ are sequences indexed over the discrete range $[q, n]$ and $m \ge 0$ is the depth of nesting (Haddad, 2021). For the special case where all sequences coincide, $a_{(i);N} = a_N$, this simplifies to

$$\hat{P}_{m, q, n}(a) = \sum_{q \le N_1 < \cdots < N_m \le n} a_{N_1}\cdots a_{N_m},$$

which enumerates products of sequences over strictly nested index ranges—forming a "triangular" domain.

The summation domain $\{(N_1, ..., N_m): q \le N_1 < ... < N_m \le n\}$ naturally reflects the combinatorics of integer partitions and set partitions: blocks of identical summation indices correspond to parts in a partition of $m$. Each partition $y_k = (y_{k,1}, ..., y_{k,m})$ with $\sum_{i=1}^m i \, y_{k,i} = m$ encapsulates how many "blocks of size $i$" appear in the indexing. This indexing underlies a powerful reduction that expresses these nested sums in terms of classical (non-nested) power sums.

2. Reduction Theorems and Partition Identities

A central result is the reduction of nested sums to explicit finite sums over integer partitions. The principal theorem [(Haddad, 2021), Theorem 3.1] asserts: $$\sum_{q \le N_1 < \cdots < N_m \le n} a_{N_1}\cdots a_{N_m} = (-1)^m \sum_{k=1}^{p(m)} \prod_{i=1}^m \frac{(-1)^{y_{k,i}}}{y_{k,i}! \, i^{y_{k,i}}} \left( \sum_{N=q}^n (a_N)^i \right)^{y_{k,i}},$$ where the sum is over all partitions $y_k$ of $m$. This result allows recursive, combinatorial, and analytic manipulations, relating nested sums to standard power sums counted according to the structure of the indexing partition.

Furthermore, closed forms for sums restricted to partitions with even or odd total number of parts are derived. For instance,

$$\sum_{\substack{\sum_i i\,y_{k,i}=m \ \sum_i y_{k,i} \text{ even}}} \prod_{i=1}^m \frac{1}{i^{y_{k,i}} y_{k,i}!} = \begin{cases} 1 & m=0, \ 0 & m=1, \ \tfrac12 & m \ge 2, \end{cases}$$

and a similar formula for the odd case [(Haddad, 2021), Theorem 6.2]. These results underpin reduction theorems used throughout the theory for explicit computations as well as for deriving classical binomial and partition identities.

3. Triangular Numbers, Partition Shifts, and Convolution Identities

Triangular numbers, $T_k = \binom{k+1}{2} = 1 + 2 + \dots + k$, play a prominent role in partition theory. For instance, if $p(n)$ denotes the number of unrestricted partitions of $n$ and $p(k,n)$ the number of partitions with exactly $k$ distinct part sizes, the identity (Hussein, 2018)

$$p\bigl(k, n + T_k\bigr) = \sum_{m=0}^{n} p(m)\,p(n-m), \qquad k \ge n,$$

relates the count of partitions with $k$ distinct magnitudes and a triangular shift to the self-convolution of $p(n)$. Analytically, this follows from previous identities of Merca expressing $p(k, n)$ in terms of partitions into exactly $j$ distinct parts and their indicator functions, and combinatorially by explicit bijection between restricted and unrestricted partitions with the shift by $T_k$ ensuring a minimal structure of $k$ baseline distinct parts.

Table: Triangular Partition Convolution Identities

Statistic	Formula	Commentary
$p(k, n + T_k)$	$\sum_{m=0}^n p(m) p(n-m)$	Shifts by $T_k$ induce convolution structure
$q(k, n)$ (distinct parts, $k$)	$p_k(n - T_k)$	$p_k$ counts partitions with parts $\leq k$
$q(k, n+T_k)$, $k \geq n$	$p(n)$	Specializes the above for large $k$

This structure generalizes: convolution identities for partitions into distinct parts or with magnitude restrictions can often be realized through appropriate shifts by triangular numbers, reflecting the minimal configuration constraints imposed by the partition statistics (Hussein, 2018).

4. Partitions Into Triangular Numbers and Attainable Partition Bijections

A further perspective links attainable partitions (arising in the study of class groups of number fields) and partitions into triangular numbers (Petersen et al., 2021). Given the cyclicity index of a partition $\lambda = (n_1, ..., n_r)$ as $c(\lambda) = \sum (3-2i) n_i$, the set of partitions with $c(\lambda) = 0$ is in bijection with partitions of $m$ into triangular numbers, where $m = \frac{1}{2} \sum n_i$. The generating function is

$$\sum_{m \ge 0} z(m) q^m = \prod_{k \ge 1} \frac{1}{1 - q^{T_k}},$$

and the full generating function for attainable partitions satisfies

$$G(q) = \sum_{n \ge 0} a(n) q^n = \frac{1}{1 - q} \prod_{k=1}^\infty \frac{1}{1 - q^{k(k+1)/2}},$$

with tight asymptotic length bounds for nonzero cyclicity partitions (Petersen et al., 2021). This shows that only $O(\sqrt n)$ triangular numbers are ever involved for $n$.

5. Triangular Sums in Representation Problems and Generating Functions

Sums over partitions into triangular numbers also appear in studies of universal representations. For example, Krachun (Krachun, 2016) established that every nonnegative integer $N$ can be written as a sum of four triangular numbers, two with even indices and two with odd indices: $$N = T_{e_1} + T_{e_2} + T_{o_1} + T_{o_2}, \quad e_1, e_2 \text{ even},\, o_1, o_2 \text{ odd}.$$ These results, rooted in additive number theory, are proven by diophantine descent and parity-refinement techniques. The underlying generating functions connect with classic identities: $$\sum_{n\ge 0} q^{T_n} = \prod_{m \ge 1} \frac{1}{1-q^m} / \prod_{m \ge 1} (1 - q^{2m-1}),$$ illuminating the structure of partitions involving triangular numbers and their natural generalizations.

6. Triangular Sums, Partition Polynomials, and $q$-Stirling Numbers

Triangular sums over set partitions, standard tableaux, and combinatorial polynomials provide explicit connections to $q$-Stirling numbers of the second kind and Catalan triangles (Prasad et al., 2021). Given a partition $\lambda \vdash n$, the polynomial

$$P_\lambda(t) = \sum_{A \in \Pi_n(\lambda)} t^{v(A)},$$

where $v(A)$ is the interlacing number of set partition $A$, serves as a key summand. The $q$-Stirling numbers are then represented as

$$S_t(n, k) = \sum_{\substack{\lambda \vdash n \ \ell(\lambda) = k}} P_\lambda(t).$$

When $t = 1$, $P_\lambda(1)$ counts set partitions of block structure $\lambda$; for $t = 0$, the number of standard tableaux of shape $\lambda$; and for $t = -1$, standard shifted tableaux. For partitions with parts at most 2, $P_\lambda(t)$ yields entries in the Catalan triangle (e.g., Touchard-Riordan polynomials).

7. Triangular Sums in Spectral Partition Problems

Beyond purely combinatorial settings, triangular sums also inform the study of spectral minimal partitions—specifically, the sum or maximum of Dirichlet-Laplacian eigenvalues over $k$-partitions of an equilateral triangle (Bogosel et al., 2017). Here, the combinatorial geometry of triangles underlies the structure of optimal partitions. The minimal average or sum

$$L_{k,1}(\Omega) = \min_{D \in P_k(\Omega)} \frac{1}{k} \sum_{i=1}^k \lambda_1(D_i)$$

adopts a spectral partitioning approach, with phase-field relaxation and assembly of geometric candidates yielding explicit minimizers for small $k$. Symmetry, regularity, and the triangular partitioning geometry are crucial for both theoretical and computational results in this context.

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These perspectives collectively illustrate that triangular sums over partitions are a unifying theme linking deep combinatorial identities, analytic generating functions, number-theoretic representation theorems, algebraic polynomial families, and even variational problems in geometry and physics. Consequences include closed formulas for multiple zeta values, refined partition statistics, generating functions for partition types with structural constraints, and algorithms or recurrences for explicit enumeration and optimization.