E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math. 5 Z. 1961 , 657#665.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# k (n, d) To count fixed density necklaces we let N(n 0 , n 1 , n k 1 ) denote the number of necklaces composed of n i occurrences of the symbol i for i = 0, 1, k 1. Let the density of the necklace d = n 1 n k 1 and n 0 = n d. It is known from Gilbert and Riordan [6] that N(n 0 , n 1 , n k 1 ) 1 n # j gcd(n0 , n k 1 ) #(j) n j) n 0 j) n k 1 j) 2.1) To get the number of fixed density necklaces with length n and density d, we sum over all possible values of n 1 , n 2 , n k 1 : N k (n, d) # n1 nk 1=d N(n d, n 1 ....

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), pp. 657--665.

....necklace algorithm. N k (n) 1 n # d n #(d)k n d , 2.2) P k (n) n # i=1 L k (i) 2.3) B k (n) # # # 1 2 (N k (n) k 1 2 k n 2 ) n even, 1 2 (N k (n) k (n 1) 2 ) n odd. 2.4) Proof. The equations for L k (n) N k (n) and B k (n) are proved by Gilbert and Riordan in [8]. The equation for P k (n) is proved in [2] In the analysis of our bracelet al..gorithm it will be useful to look at another way to count prenecklaces. Let P 0 k (n) count all k ary prenecklaces of length n that begin with 0. Notice that the number of k ary prenecklaces of length n beginning with ....

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), pp. 657--665.

.... 1 n X djn (d)k n=d (1) N k (n) 1 n X djn (d)k n=d (2) P k (n) n X i=1 L k (i) 3) B k (n) 8 : 1 2 (N k (n) k 1 2 k n=2 ) n even 1 2 (N k (n) k (n 1) 2 ) n odd (4) Proof: The equations for L k (n) N k (n) and B k (n) are proved by Gilbert and Riordan in [8]. The equation for P k (n) is proved in [2] 2 In the analysis of our bracelet al..gorithm it will be useful to look at another way to count pre necklaces. In particular, we wish to count k ary pre necklaces strictly with pre necklaces that begin with 0. Let P 0 k (n) count all k ary pre necklaces ....

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Mathematics, 5 (1961) 657-665.

....more complicated. The numbers Z(n) sequence A31) in the first column are also familiar as the number of binary irreducible polynomials of degree dividing n, and the number of n bead necklaces formed with beads of two colors, when the necklaces may not be turned over (cf. Be68, Chap. 4] GR61] MS77, Chap. 4] St99, On Single Deletion Correcting Codes 11 Problem 7.112] Fredricksen [Fr70] shows that Z(n) 1 is the number of 1 s in the truth table defining the lexicographically least de Bruijn cycle. Proof. Note that sequence A16 appears in two places in the table, for CCR ....

.... and the CCR shift register sequences Furthermore, why is V T 1 (n) sequence A48 in [EIS] equal to the number of (n 1) bead necklaces with beads of two colors and primitive period n 1, when the two colors may be interchanged but the necklaces may not be turned over (cf. Fi58] GR61] This is also the number of irreducible polynomials over F 2 of degree n 1 in which the coe#cient of x n is 1 [Car52] CMRSS] 4. Locally transitive tournaments The entry for A16 in [EIS] also indicates that this sequence arose in Brouwer s enumeration [Br80] of locally transitive ....

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657--665.

....k (i) P k (n) n X i=1 L k (i) 6) Proof: The equations for L k (n) and N k (n) are well known; a proof of the former may be found in Graham, Knuth, Patashnik (pg. 141) 11] and of the latter in Lothaire (pg. 9) 16] The equations for N k (n) and L k (n) are from Gilbert and Riordan [8]. The equation for P k (n) follows from Lemma 5 and the equation for P k (n) follows from Lemma 7. 2 In the case where k = 2 and the substring 00 is forbidden, formulae analagous to N 2 (n) L 2 (n) and P 2 (n) may be derived. Replace 2 n=d by F n=d Gamma1 F n=d 1 , where F n is the n th ....

....the same equivalence class. As representative we choose the lexicographically smallest string in the equivalence class, giving rise to the set N 2 (n) In section 1 we gave explicit expressions to count N k (n) L k (n) and P k (n) The first two enumerations were given by Gilbert and Riordan [8]. The enumeration for P k (n) follows from Lemma 7. 8 Lemma 6 If ff = a 1 Delta Delta Delta a n 2 N, then ff t 2 N for t 1. Proof: Let fi = b 1 Delta Delta Delta b n be equivalent to ff under permutation of its symbols. Then by definition of an unlabeled necklace, fi must be ....

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Mathematics, 5 (1961) 657-665.

....this transformation, we can conclude: In the new basis the B formula has nonzero coefficients only in terms where [1] occurs an odd number of times. This reduces the number of commutators approximately to the half. Exact formulas counting the number of commutators of this type can be found in [6] (formulated in the language of colored beads in a necklage) Below is the result of the change of basis. e1 = basis(z,1) e1 = 1] e2 = basis(z,2) e2 = 2] znew = eval(z,e1 e2,e1 e2) znew = 2 [1] 1,2] 1 3 [2, 1,2] 1 12 [1, 1, 1,2] 1 12 [2, 2, 1,2] 7 180 [2, 1, 1, 1,2] ....

E. N. Gilbert and J. Riordan. Symmetry types of periodic sequences. Illinois J. Math., 5:657--665, 1961.

....we let N(n 0 ; n 1 ; Delta Delta Delta n k Gamma1 ) denote the number of necklaces composed of n i occurrences of the symbol i, for i = 0; 1; k 1. Let the density of the necklace d = n 1 Delta Delta Delta n k Gamma1 and n 0 = n Gamma d. It is known from Gilbert and Riordan [6] that N(n 0 ; n 1 ; n k Gamma1 ) 1 n X jngcd(n 0 ;n 1 ; n k Gamma1 ) OE(j) n=j) n 0 =j) n 1 =j) Delta Delta Delta (n k Gamma1 =j) 1) To get the number of fixed density necklaces with length n and density d, we sum over all possible procedure Gen ( t, p : integer ) ....

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Mathematics, 5 (1961) 657-665.

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E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math. 5 Z. 1961 , 657#665.

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E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), pp. 657--665.

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E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), pp. 657--665.

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E.N. Gilbert and J. Riordan, Symmetry Types of Periodic Sequences, Illinois J. Mathematics, 657-665.

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