| Abstract | Matrice s alternirajućim predznakom na prirodan način se pojavljuju evaluacijom determinante kondenzacijskom metodom. Slutnja o enumeraciji ovih matrica (engl. The Alternating Sign Matrix Conjecture) riješena je na više načina i potaknula nova područja u istraživanju. Matrice s alternirajućim predznakom su matrice s elementima -1, 0 i 1 čiji nenul elementi alterniraju u predznaku po recima i stupcima, pri čemu je suma svakog retka i svakog stupca jednaka 1. Promatramo matrice s alternirajućim predznakom koje izbjegavaju permutacijski uzorak i zadovoljavaju svojstva temeljena na unutarnjim simetrijama. Uvodimo nove familije takvih matrica i dokazujemo njihova istaknuta kombinatorna i geometrijska svojstva, uključujući njihovu rekurzivnu prirodu. Prikazujemo profinjene enumeracije obzirom na broj specijalnih elemenata u matrici i obzirom na poziciju jedinice u prvom retku. Dokazujemo da su C -matrice s alternirajućim predznakom reda n u 1 : 1 korespondenciji s izbjegavajućim permutacijama duljine n - 1. Uspostavljena je bijekcija C -matrica i Dyckovih putova. Stasheffov politop definiran je svojom incidencijskom strukturom što otvara pitanje realizacije ovog politopa u euklidskom prostoru. Pronašli smo realizacije asociedra i drugih konveksnih politopa, u punoj općenitosti, putem familija matrica s alternirajućim predznakom. |
| Abstract (english) | The alternating sign matrices appear naturally by evaluating the determinant using the condensation method. The conjecture about the enumeration of these matrices, known as The Alternating Sign Matrix Conjecture, was solved in several ways and initiated new areas of research. The alternating sign matrices are matrices with elements -1, 0, and 1 whose non-zero elements alternate in sign by rows and columns, where the sum of each row and each column is equal to 1. We consider alternating sign matrices that avoid the permutation pattern and satisfy properties based on internal symmetries. We introduce new families of such matrices and prove their salient combinatorial and geometric properties, including their recursive nature. Vertically symmetric alternating sign matrices have the property that each rightmost 1 is located to the right of each leftmost 1. This property, applied only to adjacent rows, gives a family of \(\mathscr{C}\)-matrices that possess prominent regularities. We show refined enumerations with respect to the number of special elements in the matrix and with respect to the position of the 1s in the first row. In particular, we emphasize just some of these refined enumerations. The number \(E_{n,k}\) of the alternating sign matrices of the family \(\mathscr{C}\) of the order \(n\) whose 1 in the first row is in the \(k\)-th column is \(E_{n,k}= \frac{1}{k(n-k)} \binom{2(k-1)}{k-1} \binom{2(n-k-1)}{n-k-1}\). The above statement is proved using the recursive property of the family \(\mathscr{C}\). Furthmore, the number \(E^{-}_{n,k}\) of the \(\mathscr{C}\)-matrices of the order \(n\) which have \(k\) special elements is \(E^{-}_{n,k}=N(n-1, k+1) \) where \(N(n,k)= \frac{1}{k} \binom{n-1}{k-1} \binom{n}{k-1}\) is Narayana number. This statement is proved by showing that the numbers \(E^{-}_{n,k}\) follow the same recurrence as Narayana numbers. We prove that \(\mathscr{C}\)-matrices of order \(n\) are in 1 : 1 correspondence with avoiding permutations of the length \(n-1\). The bijection of \(\mathscr{C}\)-matrices and Dyck paths is established. Stasheff polytope is defined by its incidence structure, which raises the question of the realization of this polytope in Euclidean space. We have found realizations of the associahedron and other convex polytopes, in full generality, using families of alternating sign matrices. |
