Generalization of Some Problems with *s*-Separation

ABSTRACT

In this article we apply and discuss El-Desouky technique to derive a generalization of the problem of selecting*k* balls from an *n*-line with no two adjacent balls being *s*-separation. We solve the problem in which the separation of the adjacent elements is not having odd and even separation. Also we enumerate the number of ways of selecting k objects from *n*-line objects with no two adjacent being of separations *m*, *m + 1*, …, pm, where *p* is positive integer. Moreover we discuss some applications on these problems.

In this article we apply and discuss El-Desouky technique to derive a generalization of the problem of selecting

Cite this paper

El-Desouky, B. , Gad, M. and El-Eraqy, S. (2015) Generalization of Some Problems with*s*-Separation. *Applied Mathematics*, **6**, 1-6. doi: 10.4236/am.2015.61001.

El-Desouky, B. , Gad, M. and El-Eraqy, S. (2015) Generalization of Some Problems with

References

[1] Kplansky, I. (1943) Solution of the “Problems des Ménages”. Bulletin of the American Mathematical Society, 49, 784-785.

http://dx.doi.org/10.1090/S0002-9904-1943-08035-4

[2] Riordan, J. (1958) An Introduction to Combinatorial Analysis. Wiley, New York.

[3] Moser, W.O.J. (1986) The Number of Subsets without a Fixed Circular Distance. Journal of Combinatorial Theory, Series A, 43, 130-132.

http://dx.doi.org/10.1016/0097-3165(86)90030-0

[4] El-Desouky, B.S. (1988) On Selecting k Balls from an n-Line without Unit Separation. Indian Journal of Pure and Applied Mathematics, 19, 145-148.

[5] El-Desouky, B.S. (1988) Selecting k Balls without s-Separation. The 23rd Annual Conference on Statistics, Computer Science, Operations Research and Mathematics, Cairo, December 1988, 40-46.

[6] Mansour, T. and Sun, Y.D. (2008) On Selecting the Number of Combinations without Certain Separations. European Journal of Combinatorics, 29, 1200-1206.

http://dx.doi.org/10.1016/j.ejc.2007.06.024

[7] Mansour, T. (2014) Set Partitions with Circular Successions. European Journal of Combinatorics, 41, 207-216.

http://dx.doi.org/10.1016/j.ejc.2014.06.008

[8] Gourden, J.P. and Jackson, D.M. (1993) Combinatorial Enumeration. Wiley, New York.

[9] Pease, R.W. (1975) General Solution to the Occupancy Problem with Variably Sized Runs of Adjacent Cells Occupied by Single Balls. Mathematics Magazine, 48, 131-134.

http://dx.doi.org/10.2307/2689693

[10] Maosen, J. (1995) On Selecting k Balls from an n-Line or n-Circle without t-Separations. Northeastern Mathematical Journal, 11, 355-364.

[1] Kplansky, I. (1943) Solution of the “Problems des Ménages”. Bulletin of the American Mathematical Society, 49, 784-785.

http://dx.doi.org/10.1090/S0002-9904-1943-08035-4

[2] Riordan, J. (1958) An Introduction to Combinatorial Analysis. Wiley, New York.

[3] Moser, W.O.J. (1986) The Number of Subsets without a Fixed Circular Distance. Journal of Combinatorial Theory, Series A, 43, 130-132.

http://dx.doi.org/10.1016/0097-3165(86)90030-0

[4] El-Desouky, B.S. (1988) On Selecting k Balls from an n-Line without Unit Separation. Indian Journal of Pure and Applied Mathematics, 19, 145-148.

[5] El-Desouky, B.S. (1988) Selecting k Balls without s-Separation. The 23rd Annual Conference on Statistics, Computer Science, Operations Research and Mathematics, Cairo, December 1988, 40-46.

[6] Mansour, T. and Sun, Y.D. (2008) On Selecting the Number of Combinations without Certain Separations. European Journal of Combinatorics, 29, 1200-1206.

http://dx.doi.org/10.1016/j.ejc.2007.06.024

[7] Mansour, T. (2014) Set Partitions with Circular Successions. European Journal of Combinatorics, 41, 207-216.

http://dx.doi.org/10.1016/j.ejc.2014.06.008

[8] Gourden, J.P. and Jackson, D.M. (1993) Combinatorial Enumeration. Wiley, New York.

[9] Pease, R.W. (1975) General Solution to the Occupancy Problem with Variably Sized Runs of Adjacent Cells Occupied by Single Balls. Mathematics Magazine, 48, 131-134.

http://dx.doi.org/10.2307/2689693

[10] Maosen, J. (1995) On Selecting k Balls from an n-Line or n-Circle without t-Separations. Northeastern Mathematical Journal, 11, 355-364.