A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients

Konvalina, John

doi:10.1080/00029890.2000.12005290

Cited by 43 publications

(36 citation statements)

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“…Let F p denote the finite field with p elements where p is prime, and F n p be the n-dimensional vector space over this field. The number of the m-dimensional linear subspaces of F n p is n m p which is called Gaussian coefficient, see [3], [13] for more details. Note that to obtain a mdimensional subspace, it is sufficient to choose m linear independent vectors.…”

Section: Introductionmentioning

confidence: 99%

Projections in vector spaces over finite fields

Chen¹

2018

Ann. Acad. Sci. Fenn. Math.

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Section: Introductionmentioning

confidence: 99%

Projections in vector spaces over finite fields

Chen¹

2018

Ann. Acad. Sci. Fenn. Math.

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“…The goal was to find a high performance model developed with the fewest possible independent variables. The independent variables were combined using the Multinomial Coefficient (C n,k ), which is a generalization of the Binomial Coefficient [31,32], and which allows to select subsets of k elements from a set of n elements without repetitions:…”

Section: Ann Modelsmentioning

confidence: 99%

Soil Heat Flux Modeling Using Artificial Neural Networks and Multispectral Airborne Remote Sensing Imagery

Canelon

Chávez

2011

Remote Sensing

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Abstract:The estimation of spatially distributed crop water use or evapotranspiration (ET) can be achieved using the energy balance for land surface algorithm and multispectral imagery obtained from remote sensing sensors mounted on air-or space-borne platforms. In the energy balance model, net radiation (R n ) is well estimated using remote sensing; however, the estimation of soil heat flux (G) has had mixed results. Therefore, there is the need to improve the model to estimate soil heat flux and thus improve the efficiency of the energy balance method based on remote sensing inputs. In this study, modeling of airborne remote sensing-based soil heat flux was performed using Artificial Neural Networks (ANN). Soil heat flux was modeled using selected measured data from soybean and corn crop covers in Central Iowa, U.S.A. where measured values were obtained with soil heat flux plate sensors. Results in the modeling of G indicated that the combination R n with air temperature (T air ) and crop height (h c ) better reproduced measured values when three independent variables were considered. The combination of R n with leaf area index (LAI) from remote sensing, and R n with surface aerodynamic resistance (r ah ) yielded relative larger overall correlation coefficient values when two independent variables were included using ANN. In addition, air temperature (T air ) may be a key variable in the modeling of G as suggested by the ANN application (r of 0.83). Therefore, it is suggested that R n , LAI, r ah and h c and potentially T air be considered in future modeling studies of G. OPEN ACCESSRemote Sens. 2011, 3 1628

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“…The above Observation 3 provides us with the new combinatorial interpretation of the class of all classical F -nomial coefficients including distinguished binomial or distinguished Gauss q-binomial ones or Konvalina generalized binomial coefficients of the first and of the second kind [11,12]-which include Stirling numbers too. Naturally incidence coefficients of any reduced incidence algebra of full binomial type [8] are determined by cobweb admissible sequences therefore, they are now independently given a new cobweb combinatorial interpretation via Observation 3.…”

Section: Definitionmentioning

confidence: 99%

First Observations on Prefab Posets’ Whitney Numbers

Kwaśniewski¹

2007

AACA

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We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers' triangular array -are then calculated and the explicit formula for them is provided. Next -in the second construction -we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F -sequence independent while the second partial order is F -sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F -determined cobweb poset's Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section. (2000). 05C20, 11C08, 17B56. Mathematics Subject Classification

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A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients

Cited by 43 publications

References 4 publications

Projections in vector spaces over finite fields

Projections in vector spaces over finite fields

Soil Heat Flux Modeling Using Artificial Neural Networks and Multispectral Airborne Remote Sensing Imagery

First Observations on Prefab Posets’ Whitney Numbers

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