### Table 1 gives a few values of the number of simple permutations, 2-permutations, and 3-permutations. The sequences for simple or 3-permutations do not appear in the OEIS.

"... In PAGE 10: ... Table1 : Number of simple permutations, 2-permutations, and 3-permutations in Sn Finally, our counting results may also be used to infer the distribution of those rearrange- ment distances based on the cycle graph. For instance, Christie [2] generalised transpositions, which exchange contiguous intervals in a permutation, to the case where the exchanged in- tervals need not be contiguous, resulting in an operation called a block-interchange.... ..."

### Table 5: Permutation table.

2003

"... In PAGE 14: ... For these strings we constructed a permutation table with an advice on what to do with a certain six character string, and thus where to place the unit even when individual metrics may indicate differently. We give an impression of this in Table5 . Note that this table contains potentially BFBI BP BJBEBL entries, since the order of the strings matters.... In PAGE 16: ...reduce the potential 729 different outcomes of Table5 . Note that we use actual values that are context-specific for our running example.... ..."

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### Table 7: Index Permutation Input Pairs [1]

2007

"... In PAGE 6: ...able 6: Index Permutation (5, 4, 7, 9, 3, 8, 1, 0, 2, 6) [1]................................................... 25 Table7 : Index Permutation Input Pairs [1].... In PAGE 25: ... Cipher Rotor C0 C1 C2 C3 C4 Index Rotor Outputs (1,2) (3,4) (5,6) (7,8) (9,0) Index Rotor Inputs (6,8) (4,1) (0,9) (2,5) (3,7) Control Rotor Count 10 4 1 4 7 Table 6: Index Permutation (5, 4, 7, 9, 3, 8, 1, 0, 2, 6) [1] If we assume that the control rotors generate random permutations, the expected number of steps for a given cipher rotor i depends only on the number control rotor output letters that feed into cipher rotor Ci. The list of all 45 input pairs and their corresponding number of letters is show in Table7 . If a sufficient amount of plaintext is known, we can obtain information related to the Count column of Table 7 for each cipher rotor based on a count of the number of times that cipher rotor i has stepped.... In PAGE 25: ... The list of all 45 input pairs and their corresponding number of letters is show in Table 7. If a sufficient amount of plaintext is known, we can obtain information related to the Count column of Table7 for each cipher rotor based on a count of the number of times that cipher rotor i has stepped. From this count, we can make restrictions on the index permutation using the Pairs column.... In PAGE 26: ...9086957 Table 8: Cipher Rotor Stepping Ratios [1] Now given the cipher rotor stepping counts from Phase 1, we can use Table 8 to determine the most likely pairs of control rotor output letters for each cipher rotor. Since these are connected to the index permutation, this information combined with the information from Table7 will reduce the number of possible index permutations. A valid index permutation must contain five pairs from Table 7 such that two conditions are met.... In PAGE 26: ... Since these are connected to the index permutation, this information combined with the information from Table 7 will reduce the number of possible index permutations. A valid index permutation must contain five pairs from Table7 such that two conditions are met. The first condition is that each digit is used once and only once.... In PAGE 27: ...91 Table 9: Example Stepping Ratios [1] Using the results from Table 8, the most likely number of letters connected to cipher rotors C0, C1, C2, C3, C4 are 1, 2, 5, 7, and 11 respectively. Of the 2148 valid combination of pairs from Table7 , there are six sets of pairs that are consistent with the number of letters derived from Table 8. These six sets of pairs are shown in Table 10.... In PAGE 28: ...ith a 0.8216 probability of success. We have more information available to us from Table 11. If at any point, only one of the five cipher rotors step, then we can eliminate rows 1, 2, and 3 from Table7 since the control rotor permutation always have four active outputs. From Table 12, we see that a single rotor stepping is a relatively rare occurrence.... ..."

### Table 4: Index Permutation Input Pairs

"... In PAGE 15: ... All of the 45 possible input pairs and the corresponding number of control rotor output letters are tabulated in Table 4. If we have sufficient known plaintext available, we obtain information related to the count column of Table4 for each cipher rotor, simply based on a count of the number of times that cipher rotor i steps. Then from the pairs column, we obtain restrictions on the index permutation.... In PAGE 16: ...Table 4: Index Permutation Input Pairs compute these ratios, we assume all outputs of the control rotors are equally likely and we generate all parenleftbig26 4 parenrightbig = 14,950 of these equally likely outputs, counting the number of times that at least one element of each of the pairs in Table4 occurs. The resulting stepping ratios are given in Table 5, where step ratio is obtained by dividing step count by 14,950.... In PAGE 16: ...908696 Table 5: Cipher Rotor Stepping Ratios Now given putative cipher rotor stepping counts (as determined from known plain- text in phase one of the attack), the numbers in Table 5 can be used to determine the most likely pairs of control rotor output letters connected to each cipher rotor. Since these connections occur via the index permutation, combining this information with Table4 significantly reduces the number of possible index permutations.... In PAGE 17: ...Table4 , where 0, 1, .... In PAGE 17: ...otors C0, C1, C2, C3, C4 had stepping ratios of 0.15, 0.29, 0.60, 0.74, 0.91, respectively. According to Table 5, these results indicate that the number of letters connected to cipher rotors C0, C1, C2, C3, C4 are, most likely, 1, 2, 5, 7, 11, respectively. Six of the 2148 valid combinations of five pairs derived from Table4 are consistent with this ordering. These consistent sets of pairs appear in Table 6.... In PAGE 18: ...robability of about 0.82. There is more information available than we have used to compute the numbers in Table 7. For example, if at any point, only one cipher rotor steps, we can immediately eliminate rows 1, 2 and 3 from Table4 , since there are always four active outputs from the control permutation. Although a single rotor stepping is a relatively rare event (occurring about 2.... ..."

### Table 2 Permutations of an 838 network.

"... In PAGE 11: ... For instance, let us consider a permutation in an 838 network, as shown in Table 2. The conflict graph for the permutation in Table2 can be 11 Table 3 Comparison of maximum numbers of independent Stage Size Nodes Rounds LB Estimate Number of Subsets with Deg-desc Number of Subsets with GA Number Subsets with 3 8 100 2.56 2.... ..."

### Table 1: Number of permutations original new

"... In PAGE 11: ....3.1 Algorithmic Complexity The overall complexity of the general transformation method is given by the total number of permutations that the function is evaluated for. It can be determined for both variants of the method according to Table1 , depending on the number of -cuts (m + 1) and the number of uncertain parameters n. In general, one can say that if the number of -cuts (m + 1) is large compared to the number of uncertain parameters, the new scheme o ers a signi cantly better complexity.... In PAGE 20: ...formation method: f2(x1; x2) = (x1 2)4 + (x2 2)2 + 0:2 1 : (19) If we would like to compute f2 using fuzzy numbers decomposed into 20 -levels, we would have to consider nperm = 20 X k=1 k2 = 2870 (20) permutations according to Table1 . Counting the total number of operations o required to execute one function evaluation in the two-dimensional space, we count o = 7.... ..."

### Table 4: The permutations PC-1 and PC-2

"... In PAGE 8: ...1, page 2). The computation of the key schedule is as follows: AF the 56 bits of C35 are permuted according to a fixed permutation PC-1 (see Table4 ). Using the notation presented at the beginning of Section 2.... In PAGE 8: ... AF using BVBC and BWBC, the computation proceeds in a 16-iteration loop (BD AK CX AK BDBI): BVCX BP C4CBCXB4BVCXA0BDB5 BWCX BP C4CBCXB4BWCXA0BDB5 C3CX BPPC-2B4BVCXBWCXB5 where each C4CBCX represents a cyclic shift to the left6 of either one or two positions, de- pending on the value of CX BM C4CBCX BP AQ shift one positionBN CX BP BDBN BEBN BLBN BDBI shift two positions, otherwise. LC-2 is another fixed permutation which reduces the length of BVCXBWCX B4BD AK CX AK BDBIB5 from 56 to 48 (see Table4 ), i.... ..."

### Table 2. Permutation

2001

"... In PAGE 3: ...2 Permutation The permutation portion of a round is simply the tranposition of the bits or the permutation of the bit positions. The permutation of Figure 1 is given in Table2 (where the numbers represent bit positions in the block, with 1 being the leftmost bit and 16 being the rightmost bit) and can be simply described as: the output i of S-box j is connected to input j of S-box i. Note that there would be no purpose for a permutation in the last round and, hence, our cipher does not have one.... ..."

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### Table 2. Permutation

2001

"... In PAGE 3: ...2 Permutation The permutation portion of a round is simply the tranposition of the bits or the permutation of the bit positions. The permutation of Figure 1 is given in Table2 (where the numbers represent bit positions in the block, with 1 being the leftmost bit and 16 being the rightmost bit) and can be simply described as: the output i of S-box j is connected to input j of S-box i. Note that there would be no purpose for a permutation in the last round and, hence, our cipher does not have one.... ..."

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### Table 1 The permutation matrix

"... In PAGE 2: ... A solution of IPMSP is represented by a matrix of neurons with m rows and n columns, where n and m are the number of jobs and the number of identical machines in parallel, respectively. Table1 shows a feasible solution to an example identical parallel machine scheduling with 2 machines and 5 jobs. The permutation matrix in Table 1 represents that job 1, 4 are assigned to machine 1 and job 2, 3, 5 are assigned to machine 2.... ..."