Latin Squares and Column Ascents

Abstract

Latin squares carry a column-ascent statistic whose interior multiples are the target values.

Definition 1.1 (Latin square).

$$\forall n\in \mathrm{Nat},\forall L\in \mathrm{Fin}\left(n\right)\to \left(\mathrm{Fin}\left(n\right)\to \mathrm{Fin}\left(n\right)\right),\mathrm{IsLatin}\left(n,L\right)\iff \left(\left(\forall i\in \mathrm{Fin}\left(n\right),\mathrm{Bijective}\left(\mathrm{row}\left(L,i\right)\right)\right)\wedge \left(\forall c\in \mathrm{Fin}\left(n\right),\mathrm{Bijective}\left(\mathrm{column}\left(L,c\right)\right)\right)\right)$$

Formalization. D5/S3/Combinatorics/LatinEulerianDefs.IsLatin (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.25100. URL: https://arxiv.org/abs/2609.25100v1.

Commentary.

A map on a finite square is Latin when every row and every column is a bijection on the symbols.

Definition 1.2 (Column ascent count).

$$\forall n\in \mathrm{Nat},\forall L\in \mathrm{Fin}\left(n\right)\to \left(\mathrm{Fin}\left(n\right)\to \mathrm{Fin}\left(n\right)\right),\forall c\in \mathrm{Fin}\left(n\right),\mathrm{colAscents}\left(n,L,c\right)=\mathrm{card}\left(\{j\in \mathrm{Nat}|j+1<n\wedge L\left(j,c\right)<L\left(j+1,c\right)\}\right)$$

Formalization. D5/S3/Combinatorics/LatinEulerianDefs.colAscents (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.25100. URL: https://arxiv.org/abs/2609.25100v1.

Commentary.

For a fixed column, count the adjacent row positions whose entries increase.

Definition 1.3 (Total column ascents).

$$\forall n\in \mathrm{Nat},\forall L\in \mathrm{Fin}\left(n\right)\to \left(\mathrm{Fin}\left(n\right)\to \mathrm{Fin}\left(n\right)\right),\mathrm{totalAscents}\left(n,L\right)=\sum c\in \mathrm{Fin}\left(n\right)\mathrm{colAscents}\left(n,L,c\right)$$

Formalization. D5/S3/Combinatorics/LatinEulerianDefs.totalAscents (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.25100. URL: https://arxiv.org/abs/2609.25100v1.

Commentary.

The total ascent number is the sum of the column ascent counts over all columns.

Definition 1.4 (Interior multiples).

$$claim\iff \left(\forall n\in \mathrm{Nat},5\le n\Rightarrow \left(\forall k\in \mathrm{Nat},\left(2\le k\wedge k\le n-3\right)\Rightarrow \left(\exists L\in \mathrm{Fin}\left(n\right)\to \left(\mathrm{Fin}\left(n\right)\to \mathrm{Fin}\left(n\right)\right),\mathrm{IsLatin}\left(n,L\right)\wedge \mathrm{totalAscents}\left(n,L\right)=k\cdot n\right)\right)\right)$$

Formalization. D5/S3/Combinatorics/LatinEulerianDefs.claim (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.25100. URL: https://arxiv.org/abs/2609.25100v1.

Commentary.

For every order at least five and every integer k from two through n minus three, an order-n Latin square has total ascent number k n.

References

• Truth anchor: D5/S3/Combinatorics/LatinEulerianDefs.IsLatin
• Truth anchor: D5/S3/Combinatorics/LatinEulerianDefs.claim
• Truth anchor: D5/S3/Combinatorics/LatinEulerianDefs.colAscents
• Truth anchor: D5/S3/Combinatorics/LatinEulerianDefs.totalAscents