Some further theoretic properties of the scheme called non-deranged permutation Group, especially in relation to ascent block were identified and studied in this paper. This was done first through some computations on this scheme using prime numbers . A recursion formula for generating maximum number of block and minimum number of block were developed and it’s also observed that is equidistributed with for any arbitrary permutation group and it in decreasing order for non-deranged permutations it also established that the number of ascent block in is .

Ascent Number, Ascent set ,Ascent block and Γ1-non deranged permutations

[1] K.O. Aremu, A.H Ibrahim., S.Buoro and F.A.Akinola, Pattern Popularity in -non deranged permutations: An Algebraic and Algorithmic Approach. Annals. Computer Science Series15(2) (2017)115-122.
[2] K. O. Aremu, O. Ejima, and M. S. Abdullahi, On the fuzzy non deranged permutation Group Asian Journal ofMathematics and Computer Research, 18(4) (2017),152-157

[3] B. Clarke, A note on some Mahonian statistics, sem. Lothar.combin.53 (2005),Aricle B53a
[4] L. Euler, Institutiones Calculi differntialis in “opera omnia series prime “ Volx, (1913),Teubner,Leipzig.
[5] A.I. Garba and A.A. Ibrahim, A New Method of Constructing a Variety of Finite Group Based on Some Succession Scheme. International Journal of Physical Sciences 2(3) (2010),23-26.
[6] A.I. Garba, O. Ejima, K.O. Aremu and U. Hamisu, Non standard Young tableaux of -non deranged permutation group . Global Journal of Mathematical Analysis5(1) (2017), 21-23.
[7] G.N Han, Une transformation fondamentale sur les rearrangements de mots,Adv. Math. 105 (1994),26-41
[8] I. Haglund and L. Steven , An extension of the Foata map to standard Young tableaux, Sem. Lothar.Combin. 56 (2006),Article B56c
[9] A.A. Ibrahim, O. Ejima and K.O. Aremu, On the Representation of -non deranged permutation group Advance in Pure Mathematics, 6(2016),608-614.
[10] M. Ibrahim, A.A. Ibrahim, A.I. Garba and K.O. Aremu, Ascent on -non deranged permutation group International journal of science for global sustainability, 4(2) (2017), 27-32.
[11]M. Ibrahim and A.I. Garba Exedance on -non deranged permutations proceedings of Annual National Conference of Mathematical Association of Nigeria (MAN) , (2018), 197-201.
[12]M. Ibrahim and A.I. Garba, Descent on -non deranged permutation group Journal of Mathematical Association of Nigeria ABACUS , 46(1) (2019),12-18.
[13] P.A. MacMahon ,.Combinatory Analysis Vol. 1 and 2 (1915), Cambridge University Press(reprinted by Chesea,New York,1955)
[14] B. Sagan, A maj statistics for set partitions European J. Combin. 2 (1991), 69-79
[15]R. Simion and D. Stanton, Specialization of generalized laguerre polynomials SIAM J. Math. Anal. 25(2) (1994), 712-719
[16] R. Simion and D. Stanton, Octabasic Laguerre polynomials and permutation statistics, J.Comput. Appl. Math.68(1-2) (1996),297- 329
[17] A. Usman and A.A.Ibrahim, A new Generating Function for Aunu Patterns : Application in Integer Group Modulo n. Nigerian Journal of Basic and Applied Sciences 19(1) (2011), 1-4