Schmidt-Type Partition Identities

• Schmidt-Type theorems are identities in partition theory that weight selected indices instead of the total size, yielding generating functions equivalent to classical partition families.
• The framework employs weighted words over a commutative monoid with over-partitions and color assignments, unifying periodic examples with innovative non-periodic cases such as plane-partition identities.
• Extensions to block partitions linked with Eulerian polynomials illustrate how the method provides both analytic and bijective proofs, broadening the scope of Schmidt-type identities.

In partition theory, a Schmidt-type theorem is an identity in which partitions are weighted by selected indices rather than by their full size, and the resulting generating function is shown to coincide with the generating function of a more classical partition family. In the formulation developed by Konan, these identities are placed in a weighted-word framework over a commutative monoid, with over-partitions and color assignments serving as the basic combinatorial objects. This framework yields a uniform product formula, recovers several known periodic Schmidt-type theorems, produces genuinely non-periodic examples such as a plane-partition identity, and extends further to block partitions linked with Eulerian polynomials (Konan, 2022).

1. Weighted words, monoid weights, and over-partitions

Let $(\mathcal{M},+,0)$ be a commutative monoid, and fix a sequence

$S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$

The basic objects are over-partitions

$A=(A_1,A_2,\dots),$

with the usual convention that a final occurrence of a part may be over-lined. The $S$-weight of $A$ is defined by

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$

where $A_i\cdot s_i$ means that $s_i$ is added to itself $A_i$ times in the monoid (Konan, 2022).

To record refinements, one fixes a color set

$C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$

with the rule that only $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$0 may colour the monoid element $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$1, while $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$2 colour $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$3. If $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$4 carries a color-assignment

$S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$5

then its total weight is

$S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$6

This formulation isolates the role of the index set through the sequence $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$7: the combinatorial statistic is not the total size of the partition, but the sum of those $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$8 attached to indexed positions (Konan, 2022).

The conceptual significance of this framework is that it accommodates both periodic and non-periodic index patterns. This suggests that Schmidt-type identities are not confined to residue classes or periodic gap conditions, even though many classical examples arise from periodic choices of $S=(s_1,s_2,\dots), \qquad s_i\in \mathcal{M}.$9.

2. The main generating-function theorem

The central result is Theorem 1.11 of Konan. There is a bijection $A=(A_1,A_2,\dots),$0 on over-partitions sending $A=(A_1,A_2,\dots),$1, preserving the $A=(A_1,A_2,\dots),$2-weight and turning conjugation into a recolouring of parts. As a consequence,

$A=(A_1,A_2,\dots),$3

where the product runs over all $A=(A_1,A_2,\dots),$4 for which $A=(A_1,A_2,\dots),$5 in $A=(A_1,A_2,\dots),$6 (Konan, 2022).

The proof is organized around three steps. First, one defines $A=(A_1,A_2,\dots),$7 by conjugating the Ferrers diagram of $A=(A_1,A_2,\dots),$8 and then recolouring each part of size $A=(A_1,A_2,\dots),$9 by $S$0 or $S$1. Second, one proves that

$S$2

and that $S$3 is an involution. Third, a standard combinatorial decomposition of the $S$4-fixed points yields the product formula (Konan, 2022).

This theorem is the structural core of the subject in the weighted-word approach. Rather than proving each Schmidt-type identity separately, it reduces the problem to specifying a sequence $S$5 and a colour specialization. The product side then emerges uniformly, with the combinatorial content encoded in the choice of nonzero entries of $S$6.

3. Classical periodic Schmidt-type theorems

Periodic choices of $S$7 recover the classical Schmidt-type partition identities studied by Schmidt, Andrews, Paule, Uncu, and others. In these cases the nonzero terms of $S$8 occur at regular index intervals, and the general product formula specializes to familiar generating functions (Konan, 2022).

Case	Choice of $S$9	Consequence
Distinct-part Schmidt	$A$0 if $A$1 is odd, $A$2 otherwise; trivial colours	Generates ordinary partitions
Unrestricted Schmidt	Same $A$3, repetition allowed	Generates two-coloured partitions
$A$4-elongated case	$A$5 iff $A$6, $A$7 otherwise	Recovers Andrews–Paule product form

In the distinct-part case, one obtains

$A$8

and this is bijectively equal to

$A$9

the ordinary partition generating function (Konan, 2022).

If the same sequence $|A|_S=\sum_{i\ge1} A_i\cdot s_i,$0 is used but repetition is allowed, then

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$1

now interpreted as the generating function of two-coloured partitions (Konan, 2022). This is the form associated in the recent literature with Andrews–Paule and Uncu.

For $|A|_S=\sum_{i\ge1} A_i\cdot s_i,$2-elongated partitions, one chooses a period $|A|_S=\sum_{i\ge1} A_i\cdot s_i,$3 and imposes

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$4

together with periodic colours $|A|_S=\sum_{i\ge1} A_i\cdot s_i,$5. The product formula becomes

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$6

which recovers the Andrews–Paule product form and, after suitable specialization, Theorem 1.3 in that setting (Konan, 2022).

These examples show that the weighted-word framework subsumes the standard periodic-gap Schmidt-type theorems. The paper’s summary further states that it recovers not only Schmidt, Uncu/Andrews–Paule, and Andrews–Paule on $|A|_S=\sum_{i\ge1} A_i\cdot s_i,$7-elongated partitions, but also Andrews–Keith refinements (Konan, 2022).

4. Non-periodic sequences and the plane-partition example

A notable feature of the framework is that it produces new Schmidt-type theorems for non-periodic sequences. Let

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$8

and define

$|A|_S=\sum_{i\ge1} A_i\cdot s_i,$9

The contributing indices are exactly

$A_i\cdot s_i$0

(Konan, 2022).

For this choice of $A_i\cdot s_i$1,

$A_i\cdot s_i$2

which is the classical MacMahon generating function for plane partitions of weight $A_i\cdot s_i$3 (Konan, 2022). Hence the coefficient of $A_i\cdot s_i$4 in the left-hand side is the number of plane partitions of $A_i\cdot s_i$5.

Equivalently, for every non-negative integer $A_i\cdot s_i$6, the number of partitions such that

$A_i\cdot s_i$7

is equal to the number of plane partitions of $A_i\cdot s_i$8 (Konan, 2022).

This example is significant because it is explicitly non-periodic. A common misconception is that Schmidt-type partition theorems are inherently modular or residue-class phenomena. The plane-partition example shows that the essential structure is the selective indexing itself, not periodicity.

5. Block partitions and Eulerian polynomials

Konan also introduces block over-partitions, a new family generalizing the $A_i\cdot s_i$9-elongated partitions. Fix a strictly increasing sequence

$s_i$0

of positive integers, and set

$s_i$1

A block over-partition of type $s_i$2 is an over-partition

$s_i$3

such that, within each block of indices

$s_i$4

the parts $s_i$5 are all at least the next block’s minimal part. The set of such objects is denoted $s_i$6 (Konan, 2022).

The corresponding generating series is expressed in terms of block-Euler polynomials

$s_i$7

which generalize Eulerian polynomials via descent-enumeration in $s_i$8. Theorem 1.23 states that if

$s_i$9

then

$A_i$0

The proof proceeds by showing that $A_i$1 is in bijection with pairs consisting of a sequence of permutations, one in each $A_i$2, and an over-partition, followed by a blockwise Glaisher-type flat/regular argument (Konan, 2022).

Under a colour specialization in which all $A_i$3 except one grounding colour, the block-Euler polynomials reduce to the classical Eulerian polynomials $A_i$4. The block-partition generating function then becomes a product of Eulerian polynomials over linear denominators (Konan, 2022). Combinatorially, the descent structure of the permutations encoding each block corresponds to patterns of equality versus strict decrease across the block boundary, so Eulerian numbers count descents, that is, strict drops, in each block (Konan, 2022).

This extension places Schmidt-type partition identities in direct contact with permutation statistics. It also clarifies that the $A_i$5-elongated case is only one member of a larger family organized by block data rather than a single period.

6. Related bijections, refinements, and the scope of the term

Several contemporaneous papers develop Schmidt-type partition theory by direct bijection. Li and Yee characterize Schmidt type partitions in a general and refined form: for fixed $A_i$6, the number

$A_i$7

of $A_i$8-tuple strict partitions of $A_i$9 with specified lengths equals the number

$C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$0

of Schmidt $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$1-partitions of $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$2 with exactly $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$3 parts, and they also prove unrestricted and overpartition analogues (Li et al., 2022). Waldron relates Mork’s bijection to a version of Sylvester’s bijection on $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$4-modular diagrams, derives refinements of existing results, and introduces a color-conjugate map that sends partitions counted at indices $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$5 to $C=\{c_i: i\ge0\}\cup\{\bar c_i: i\ge1\},$6-colored partitions (2207.14586). Andrews and Keith treat periodic but otherwise arbitrary subsets of counted or uncounted parts, using a colored Stockhofe-type bijection to identify these families with colored partitions satisfying specified residue conditions (Andrews et al., 2022). Uncu gives a combinatorial proof of the Andrews–Paule theorem and a four-variable refinement, again centered on hooks, Ferrers diagrams, and two-color partitions (Ji, 2021).

These developments reinforce two structural themes. First, Schmidt-type theorems are often best understood as correspondences between selective index-sums and color data. Second, the same identities admit both analytic and bijective realizations. Konan’s weighted-word theorem provides a uniform generating-function mechanism, while the bijective papers isolate explicit maps such as Mork’s bijection, Sylvester-type constructions, Wright’s Durfee-square bijection, and the color-conjugate map (Konan, 2022).

The phrase “Schmidt-type theorem” is not unique to partition theory. In Diophantine approximation it appears in generalizations of Schmidt’s subspace theorem for moving hypersurfaces, closed subschemes, and algebraic points of bounded degree (Quang, 2016, Heier et al., 2017, Zhao, 12 Feb 2025). In operator theory, “Schmidt subspaces” of a Hankel operator are described as images of model spaces under isometric multipliers (Pushnitski et al., 2019). This broader usage can cause terminological ambiguity. In the partition-theoretic sense, however, a Schmidt-type theorem refers specifically to identities in which only designated indexed parts contribute to the weight, and the resulting enumerant is equated with a classical partition class or one of its colored or overpartition analogues (Konan, 2022).