Descending Plane Partitions

• Descending plane partitions are finite arrays of positive integers with row-weak and column-strict decreases, serving as key combinatorial objects equinumerous with ASMs and TSSCPPs.
• Their lattice path representations and determinant formulations enable refined enumerations and q-analog extensions, bridging algebraic and combinatorial techniques.
• They form a critical nexus in combinatorics, connecting permutation matrices, alternating sign matrices, and refined generating functions, while posing challenges for direct bijections.

A descending plane partition (DPP) is a finite array of positive integers arranged with strict combinatorial constraints, forming one of the principal classes of enumerative objects equinumerous with alternating sign matrices (ASMs) and totally symmetric self-complementary plane partitions (TSSCPPs). DPPs encapsulate deep connections among non-intersecting lattice paths, determinant and orthogonal polynomial techniques, and the theory of refined enumeration, and are central to longstanding open problems in bijective combinatorics.

1. Formal Definitions and Structure

A DPP of order $n$ is a finite array $\pi = (a_{i,j})$ of positive integers, typically presented in shifted shape, satisfying:

• Row-weak decrease: $a_{i,j} \geq a_{i,j+1}$ for all valid $i,j$.
• Column-strict decrease: $a_{i,j} > a_{i+1,j}$ for all valid $i < r$ and $j \leq \lambda_{i+1}$, where $\lambda_k$ denotes the length of row $k$.
• Diagonal/first-part and length constraints: The first part of row $i$ satisfies $\pi = (a_{i,j})$0 with initial parameter $\pi = (a_{i,j})$1.

A part $\pi = (a_{i,j})$2 is termed special if $\pi = (a_{i,j})$3; otherwise, it is nonspecial ($\pi = (a_{i,j})$4). DPPs may be further classified by order, namely the upper bound $\pi = (a_{i,j})$5 on the entries, frequently denoted $\pi = (a_{i,j})$6.

2. Lattice Path Interpretation

DPPs admit a canonical representation in terms of families of non-intersecting lattice paths in the first quadrant. Each row $\pi = (a_{i,j})$7 maps to a path $\pi = (a_{i,j})$8 from $\pi = (a_{i,j})$9 to $a_{i,j} \geq a_{i,j+1}$0, constructed via the sequence of parts in the row:

• Paths use only North $a_{i,j} \geq a_{i,j+1}$1 and West $a_{i,j} \geq a_{i,j+1}$2 steps.
• The step structure is: traverse West to just east of the horizontal line $a_{i,j} \geq a_{i,j+1}$3 for each part, then step North.

The column-strict and row-weak constraints guarantee that these path families are vertex-disjoint. This sets up a Lindström–Gessel–Viennot (LGV) correspondence between DPPs and determinants, and enables translating combinatorial questions into questions about lattice path weights and configurations (Fulmek, 2018, Behrend et al., 2011, Behrend et al., 2012).

3. Enumeration and Refined Generating Functions

Product formula

The foundational enumerative result for DPPs of order $a_{i,j} \geq a_{i,j+1}$4 is the Mills–Robbins–Rumsey–Andrews product formula: $a_{i,j} \geq a_{i,j+1}$5 This formula coincides with that for $a_{i,j} \geq a_{i,j+1}$6 ASMs, establishing a key instance of equinumerosity (Behrend et al., 2011, Francesco, 2012, Rosengren, 2012).

Determinant interpretation

The LGV framework yields

$a_{i,j} \geq a_{i,j+1}$7

and its $a_{i,j} \geq a_{i,j+1}$8-deformation underpins the $a_{i,j} \geq a_{i,j+1}$9-enumeration of DPPs (Rosengren, 2012).

Refined statistics and generating functions

Natural DPP statistics include:

• $i,j$0: number of nonspecial parts ($i,j$1)
• $i,j$2: number of special parts ($i,j$3)
• $i,j$4: number of maximal parts ($i,j$5)
• $i,j$6: number of $i,j$7's plus number of rows of length $i,j$8

The triply and quadruply refined generating functions have determinant representations: $i,j$9

$a_{i,j} > a_{i+1,j}$0

with explicit determinantal forms and bilinear relations induced by the Desnanot–Jacobi identity (Behrend et al., 2012).

4. Bijections with Permutation Matrices and Refined Equidistributions

There is a canonical bijection $a_{i,j} > a_{i+1,j}$1 between permutation matrices of size $a_{i,j} > a_{i+1,j}$2 and DPPs without special parts, $a_{i,j} > a_{i+1,j}$3. This bijection employs the inversion word of a permutation, mapping inversion sequences to DPPs with no special parts, with the explicit mapping:

• For $a_{i,j} > a_{i+1,j}$4, inversion word $a_{i,j} > a_{i+1,j}$5, form a DPP with row lengths $a_{i,j} > a_{i+1,j}$6 and entries

$a_{i,j} > a_{i+1,j}$7

This construction preserves key statistics:

• The number of parts in the DPP equals the number of inversions in the permutation.
• The number of rows of length $a_{i,j} > a_{i+1,j}$8 corresponds to specific permutation matrix orbit statistics (Fulmek, 2018, Striker, 2010, Fulmek, 2016).

This implies $a_{i,j} > a_{i+1,j}$9 and provides a refined generating function: $i < r$0

5. DPPs, ASMs, TSSCPPs, and Further Connections

Descending plane partitions form one apex of a combinatorial triangle with ASMs and TSSCPPs. All three classes are equinumerous by a nontrivial product formula, yet no explicit, direct bijection between DPPs and ASMs (or DPPs and TSSCPPs) exists for general $i < r$1 (Francesco, 2012, Aigner et al., 2021, Keller et al., 2017). Major links include:

• ASM–DPP Correspondence: Determinant representations of ASM and DPP refined generating functions are shown to coincide via explicit kernel conjugations and infinite matrix factorizations (using Izergin–Korepin and LGV techniques) (Behrend et al., 2011, Francesco, 2012). Statistics such as inversion numbers in ASMs and parts in DPPs are equidistributed via these determinantal formulas.
• Refined statistics correspondences: There exist quadruples of statistics (and, in extended frameworks, $i < r$2-tuples) that are equidistributed between ASMs and DPPs, but only in enlarged signed settings do higher-dimensional correspondences become natural (Aigner et al., 2021).
• Catalan DPPs and pattern avoidance: There is a subset of DPPs counted by Catalan numbers, corresponding via the generating tree method to 231-avoiding permutations, providing a direct link between DPPs and permutation pattern theory (Keller et al., 2017).

6. Open Problems and Bijection Obstacles

Despite determinant and generating function equivalences, as well as matching refined statistics, an explicit, natural, and direct bijection between DPPs (with all parts $i < r$3) and ASMs remains elusive (Francesco, 2012, Aigner et al., 2021). Extensions to signed objects and arrowed monotone triangles yield joint distributions matching in $i < r$4 statistics, but no sign-free combinatorial mapping has been realized. This persistent gap is conjectured to be deeply linked to the need for sign-reversing involutions and structural decorations beyond the pure DPP and ASM frameworks.

7. Generalizations and Further Directions

• $i < r$5-enumerations and weighted analogs of DPPs are accessible via orthogonal polynomial techniques and determinant evaluations (Rosengren, 2012).
• Truncated determinant techniques (infinite matrices and principal minor correspondence) yield broad classes of refined enumerations and asymptotic results (Francesco, 2012).
• Extensions to signed and set-valued objects expand the landscape, allowing for fine-grained equidistribution but increasing combinatorial complexity (Aigner et al., 2021).
• Catalan subsets and bijective growth rules remain prominent in the ongoing study of DPPs, with hopes that such structured subfamilies may ultimately illuminate the bijective landscape for the entirety of DPP, ASM, and TSSCPP classes.

DPPs thus occupy a critical position in algebraic combinatorics, with their determinant, path, and permutation encodings bridging deep enumerative equivalences and continually motivating new combinatorial and algebraic insights (Fulmek, 2018, Behrend et al., 2011, Francesco, 2012, Behrend et al., 2012, Striker, 2010, Keller et al., 2017, Aigner et al., 2021, Rosengren, 2012).