Some Counting Problems; Multinomial Coefficients, The Principle of Inclusion–Exclusion, Sylvester’s Formula, The Sieve Formula

2011 ◽  
pp. 205-255 ◽  
Author(s):  
Jean Gallier
Keyword(s):  
Counting Problems ◽  
Multinomial Coefficients

scholarly journals A computational proof of complexity of some restricted counting problems

2011 ◽  
Vol 412 (23) ◽  
pp. 2468-2485 ◽  
Author(s):  
Jin-Yi Cai ◽  
Pinyan Lu ◽  
Mingji Xia
Keyword(s):  
Counting Problems

scholarly journals On a link between Dirichlet kernels and central multinomial coefficients

2015 ◽  
Vol 338 (9) ◽  
pp. 1567-1572
Author(s):  
Michelle Rudolph-Lilith ◽  
Lyle E. Muller
Keyword(s):  
Multinomial Coefficients

Media Clips: Boulder Lottery; The Probability of Slot Machines

2013 ◽  
Vol 107 (3) ◽  
pp. 172-175
Author(s):  
Kristy B. McGowan ◽  
Nathan J. Lowe Spicer
Keyword(s):  
Slot Machines ◽  
Counting Problems ◽  
State Lottery ◽  
The Media

Students analyze items from the media to answer mathematical questions related to the article. The clips this month, from the Colorado State lottery and a Marilyn vos Savant column on probability, involve probability and counting problems.

Informing Practice: Counting Using Sets of Outcomes

2012 ◽  
Vol 18 (3) ◽  
pp. 132-135 ◽  
Author(s):  
Elise Lockwood
Keyword(s):  
Counting Problems

A branch of mathematics—combinatorics—is explored through counting problems.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

scholarly journals LOOPS, SURFACES AND GRASSMANN REPRESENTATION IN TWO- AND THREE-DIMENSIONAL ISING MODELS

1999 ◽  
Vol 14 (29) ◽  
pp. 4549-4574 ◽  
Author(s):  
C. R. GATTRINGER ◽  
S. JAIMUNGAL ◽  
G. W. SEMENOFF
Keyword(s):  
Three Dimensional ◽  
Ising Models ◽  
Algebraic Representation ◽  
Simple Derivation ◽  
Counting Problems ◽  
Intrinsic Geometry ◽  
Geometry Factor ◽  
Loop Expansion ◽  
Loop Surface

We construct an algebraic representation of the geometrical objects (loop and surface variables) dual to the spins in 2 and 3D Ising models. This algebraic calculus is simpler than dealing with the geometrical objects, in particular when analyzing geometry factors and counting problems. For the 2D case we give the corrected loop expansion of the free energy and the radius of convergence for this series. For the 3D case we give a simple derivation of the geometry factor which prevents overcounting of surfaces in the intrinsic geometry representation of the partition function, and find a classification of the surfaces to be summed over. For 2 and 3D we derive a compact formula for 2n-point functions in loop (surface) representation.

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