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A note on the determinant of a Toeplitz-Hessenberg matrix

Mircea Merca (2013)

Special Matrices

The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

A note on the exact expected length of the $k$ th part of a random partition.

Eriksson, Kimmo (2010)

Integers

A partition identity with certain applications

Hsu, L.C. (1991)

Portugaliae mathematica

A proof of the identities of Gordon and MacMahon and two new identities. (La démonstration des identités de Gordon et MacMahon et de deux identités nouvelles.)

Désarménien, Jacques (1986)

Séminaire Lotharingien de Combinatoire [electronic only]

A refinement of a partition theorem of Sellers.

Chen, Kuo-Jye (2004)

Integers

A sharp bound for the reconstruction of partitions.

Vatter, Vincent (2008)

The Electronic Journal of Combinatorics [electronic only]

Advanced determinant calculus.

Krattenthaler, Christian (1999)

Séminaire Lotharingien de Combinatoire [electronic only]

An algebraic summation over the set of partitions and some strange evaluations

Chu Wenchang (1995)

Rendiconti del Seminario Matematico della Università di Padova

An application of Pólya’s enumeration theorem to partitions of subsets of positive integers

Xiao Jun Wu, Chong-Yun Chao (2005)

Czechoslovak Mathematical Journal

Let $S$ be a non-empty subset of positive integers. A partition of a positive integer $n$ into $S$ is a finite nondecreasing sequence of positive integers $a_{1}, a_{2}, \dots, a_{r}$ in $S$ with repetitions allowed such that $\sum_{i = 1}^{r} a_{i} = n$ . Here we apply Pólya’s enumeration theorem to find the number $¶ (n; S)$ of partitions of $n$ into $S$ , and the number $D P (n; S)$ of distinct partitions of $n$ into $S$ . We also present recursive formulas for computing $¶ (n; S)$ and $D P (n; S)$ .