The aim of studying and bibliography

The main aim of study: Acquainting students with selected topics in higher mathematics.

The general information about the module: The topics of the classes will be chosen by students at the end of the first semester.

Bibliography required to complete the module

Bibliography used during lectures

1	Z. Palka, A. Ruciński	Wykłady z kombinatoryki. Przeliczanie	Warszawa WNT.	2004
2	V. Bryant	Aspekty kombinatoryki	Warszawa WNT.	1997
3	D. Knuth, O. Patashnik, R. L. Graham	Matematyka konkretna	Wydawnictwo Naukowe PWN.	2020

Bibliography used during classes/laboratories/others

1	Z. Palka, A. Ruciński	Wykłady z kombinatoryki. Przeliczanie	Warszawa WNT.	2004
2	D. Knuth, O. Patashnik, R. L. Graham	Matematyka konkretna	Wydawnictwo Naukowe PWN.	2020

Bibliography to self-study

1

I. Koźniewska

Równania rekurencyjne

Warszawa PWN.

1972

Basic requirements in category knowledge/skills/social competences

Formal requirements: The student satisfies the formal requirements set out in the study regulations.

Basic requirements in category knowledge: Student knows basis of combinatorics.

Basic requirements in category skills: Student knows some methods of combinatorics and their applications.

Basic requirements in category social competences: Student understands the necessity of the systematic learning.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
01	knows properties of Stirling numbers	lecture, classes, e-learning	test	K_W01+ K_W04+ K_W05+ K_W07+ K_U02+ K_U03+ K_U04+ K_K01+ K_K02+ K_K04+ K_K07+	P7S_KK P7S_KO P7S_KR P7S_UK P7S_UO P7S_UW P7S_WG
02	student knows the method of undetermined coefficients of solving linear nonhomogeneous recurrence equations	lecture, classes, e-learning	test	K_W01+ K_W04+ K_W05+ K_W07+ K_U02+ K_U03+ K_U04+ K_K01+ K_K02+ K_K04+ K_K07+	P7S_KK P7S_KO P7S_KR P7S_UK P7S_UO P7S_UW P7S_WG
03	can create a recurrence relation	lecture, classes, e-learning	test	K_W04++ K_U02++ K_U03++ K_K07++	P7S_KK P7S_KO P7S_KR P7S_UK P7S_UO P7S_UW P7S_WG
04	can solve a nonlinear equation transformable to linear recurrence relation	lecture, classes, e-learning	test	K_W04++ K_U03++ K_K07++	P7S_KK P7S_KO P7S_KR P7S_UW P7S_WG
05	can solve a system of linear recurrence equations by method of elimination	lecture, classes, e-learning	test	K_W05++ K_K07++	P7S_KK P7S_KO P7S_KR P7S_WG

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Selection schemes. Generalizations of Newton symbol. Permutations and cycles. Stirling numbers of the first and the second kind.	W01, W02, W03, C01, C02	MEK01
3	TK02	Recurrences. Linear nonhomogeneous equations with constant coefficients: method of undetermined coefficients. Generating functions and their applications.	W04, W05, W06, C03, C04	MEK02
3	TK03	Nonlinear recurrence equations transformable to linear recurrence equations.	W07, C05, C06	MEK04
3	TK04	Applications of recurrence relations in economy. Graphs and combinatorial interpretations of some numbers defined recursively.	W08, W09, C07	MEK02 MEK03
3	TK05	Some applications of recurrence relations in graph theory (counting of independent and dominating sets in some classes of graphs)	W10, W11, C09	MEK02 MEK03
3	TK06	Systems of linear recurrence equations, method of elimination. Applications of systems of linear recurrence equations,	W12, W13, C10, C11	MEK02 MEK03 MEK05
3	TK07	Homogeneous and nonhomogeneous systems of linear recurrence equations. Matrix method of solving the systems of linear recurrence equations.	W14, W15, C12, C13, C15	MEK02
3	TK08	Test.	C08, C14	MEK01 MEK02 MEK03 MEK04 MEK05

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 10.00 hours/sem.	contact hours: 30.00 hours/sem.
Class (sem. 3)	The preparation for a Class: 10.00 hours/sem. The preparation for a test: 5.00 hours/sem.	contact hours: 45.00 hours/sem.
Advice (sem. 3)	The preparation for Advice: 3.00 hours/sem.	The participation in Advice: 1.00 hours/sem.
Credit (sem. 3)	The preparation for a Credit: 10.00 hours/sem.	The written credit: 3.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	Attendance at the lecture
Class	Two written tests on the dates agreed with the students. To pass it the student have to solve the test in at least 50%. To receive credit the student must attend training classes.
The final grade	The final mark is the grade of classes.

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: yes

1	D. Bród	On Some Combinatorial Properties of Balancing Split Quaternions	2024
2	D. Bród; A. Szynal-Liana	On generalized bihyperbolic Mersenne numbers	2024
3	D. Bród; A. Szynal-Liana	A new hybrid generalization of Fibonacci and Fibonacci-Narayana polynomials	2023
4	D. Bród; A. Szynal-Liana	Generalized commutative Jacobsthal quaternions and some matrices	2023
5	D. Bród; A. Szynal-Liana	Jacobsthal numbers, Pell numbers, their generalizations and applications	2023
6	D. Bród; A. Szynal-Liana	On Bihypernomials Related to Balancing and Chebyshev Polynomials	2023
7	D. Bród; A. Szynal-Liana; I. Włoch	One-Parameter Generalization of Dual-Hyperbolic Jacobsthal Numbers	2023
8	D. Bród; A. Szynal-Liana; I. Włoch	One-parameter generalization of the bihyperbolic Jacobsthal numbers	2023
9	G. Bilgici; D. Bród	On r-Jacobsthal and r-Jacobsthal-Lucas Numbers	2023
10	D. Bród; A. Michalski	On Generalized Jacobsthal and Jacobsthal–Lucas Numbers	2022
11	D. Bród; A. Szynal-Liana	On a New Generalization of Jacobsthal Hybrid Numbers	2022
12	D. Bród; A. Szynal-Liana; I. Włoch	On some combinatorial properties of generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions	2022
13	D. Bród; A. Szynal-Liana; I. Włoch	One-parameter generalization of dual-hyperbolic Pell numbers	2022
14	D. Bród; A. Szynal-Liana; I. Włoch	Two generalizations of dual-complex Lucas-balancing numbers	2022
15	D. Bród; A. Szynal-Liana; I. Włoch	Two-parameter generalization of bihyperbolic Jacobsthal numbers	2022
16	D. Bród	On balancing quaternions and Lucas-balancing quaternions	2021
17	D. Bród	On trees with unique locating kernels	2021
18	D. Bród; A. Szynal-Liana; I. Włoch	Balancing hybrid numbers, their properties and some identities	2021
19	D. Bród; A. Szynal-Liana; I. Włoch	Bihyperbolic numbers of the Fibonacci type and their idempotent representation	2021
20	D. Bród; A. Szynal-Liana; I. Włoch	On a new generalization of bihyperbolic Pell numbers	2021
21	D. Bród; A. Szynal-Liana; I. Włoch	On a new one-parameter generalization of dual-complex Jacobsthal numbers	2021
22	D. Bród; A. Szynal-Liana; I. Włoch	On a new two-parameter generalization of dual-hyperbolic Jacobsthal numbers	2021
23	D. Bród; A. Szynal-Liana; I. Włoch	On some combinatorial properties of bihyperbolic numbers of the Fibonacci type	2021
24	D. Bród; A. Włoch	(2,k)-Distance Fibonacci Polynomials	2021
25	D. Bród	On a new Jacobsthal-type sequence	2020
26	D. Bród	On distance (k, t)-Fibonacci numbers and their applications	2020
27	D. Bród	On some properties of split Horadam quaternions	2020
28	D. Bród	On split r-Jacobsthal quaternions	2020
29	D. Bród; A. Szynal-Liana	On J(r,n)-Jacobsthal Hybrid Numbers	2020
30	D. Bród; A. Szynal-Liana	On some combinatorial properties of P(r,n)-Pell quaternions	2020
31	D. Bród; A. Szynal-Liana; I. Włoch	On the combinatorial properties of bihyperbolic balancing numbers	2020
32	D. Bród; A. Szynal-Liana; I. Włoch	Two Generalizations of Dual-Hyperbolic Balancing Numbers	2020
33	D. Bród	On a new generalization of split Pell quaternions	2019
34	D. Bród	On a new one parameter generalization of Pell numbers	2019
35	D. Bród; A. Szynal-Liana	On a new generalization of Jacobsthal quaternions and several identities involving these numbers	2019