### Table 1: Complementation, transposition, and their composition

2004

"... In PAGE 4: ...4 Also, if C denotes the operation of complementation mentioned in Remark 1, T denotes the operation of transposition (see Remark 1 again), and CT denotes the composition of C and T , which is obviously commutative, then one can use Table1 to reduce the... In PAGE 4: ... Moreover, if in the the expression for that Mm;n we switch m and n, we get the number of permutations that simultaneously avoid the patterns p3, p4 and p6, as well as that avoiding the patterns p3, p6 and p7, due to the operation of transposition and that of composition of complementation and transposition. So, using Table1 , one can divide all the possibilities into equivalence classes, and we need to consider a representative from each class. We indicate the equivalence classes and their representatives in the corresponding tables of Sections and .... ..."

Cited by 3

### Table 2. Basic performance results without transposition tables.

1985

"... In PAGE 20: ....1. Without Transposition Table By disabling the transposition table, it is easier to observe characteristics of the parallel searching algorithm that may be obscured when the table is in use. Table2 summarizes the results obtained on five ply searches without a transposition table. The nodes column corresponds to the number of leaf nodes searched by all the processors combined.... In PAGE 21: ... This reordering is the reason for the change in the move selected by systems of different processor sizes and for the occasional decrease in leaf nodes searched by the larger systems [17]. For example, Table2 shows an apparent anomaly in the reduction of leaf nodes searched when four processors are used. This was due to the biasing effect by a single test position on the average number of nodes searched.... ..."

Cited by 32

### 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 IV RESULTS OF THE PARAMETER SETTING STUDY FOR THE PETERSEN PERMUTATION REPRESENTATION WHILE MAXIMIZING NUMBER OF INDIVIDUALS REMOVED. MUTATION RATE IS THE NUMBER OF TRANSPOSITIONS USED. THE NUMBER OF SIMULATIONS PER PARAMETER SET IS N = 30.

### Table 1: Sequences of transpositions used by Algorithm 1 As an example, assume the mapping of node ! = 103 2 M(3) onto a node 2 S(4). Initially, Algorithm 1 sets = 1234. Since m[1] = 1, a (21) transposition is performed on 1234, giving 2134. Next, the algorithm examines the coordinate of ! along the 2nd dimension (m[2]). Since m[2] = 0, no transpositions are performed at this step. Next, m[3] is examined, resulting in a sequence of transpositions (4 3) (3 2) (2 1). Such a sequence a ects as shown below:

1994

"... In PAGE 9: ...et the transposition of two digits p[r], p[s] in be denoted by (r s) (i.e., (r s) corresponds to the exchange of the digits occupying the rth and the sth positions in permutation ). Table1 lists the sequences of transpositions used by Algorithm 1 along the (n?1) dimensions of the mesh. Note that if the coordinate of the mesh node along dimension i is m[i], then only the rst m[i] transpositions of the sequence corresponding... In PAGE 9: ... Proof : Without loss of generality, assume that the ith coordinates of ! and !i (respectively, m[i] and m0[i]) are such that m[i] = m0[i] ? 1. By inspection of Algorithm 1 and Table1 , we note that the mapping of ! 2 V (M(n ? 1)) onto 2 V (S(n)) uses a sequence of transpositions of the form:... In PAGE 24: ... Proof : Without loss of generality, we assume that the ith coordinates of ! and !i;2 (respectively, m[i] and m00[i]) are such that m[i] = m00[i] ? 2. By inspection of Algorithm 1 and Table1 , we note that the mapping of ! 2 V (M(n ? 1)) onto 2 V (S(n)) uses a sequence of transpositions of the form: = (a b)(c d) : : :(i j)(o p) : : :(y z) Accordingly, !i;2 is mapped onto i;2 via a sequence of transpositions of the form:... In PAGE 25: ...Note that identical transpositions are used in and i;2, except for the fact that !i;2 has two transpositions more than does !i (namely, (k l) and (m n)). An inspection of Table1 reveals that these extra transpositions are ((i ? m00[i] + 3) (i ? m00[i] + 2)) and ((i ? m00[i] + 2) (i ? m00[i] + 1)), respectively. Alternatively, we can use the cyclic representation (k l)(m n) = (k l)(l n), since l = m = n + 1 = k ? 1.... ..."

Cited by 1

### Table 1: Sequences of transpositions used by Algorithm 1 The mapping algorithm described in this paper di ers slightly from that proposed in [13] in respect to the de nition of a transposition (r s). In Algorithm 1, (r s) corresponds to an exchange of the digits occupying the rth and the sth positions in permutation . In the corresponding algorithm described in [13], (r s) corresponds to an exchange of digits r and s in . Both approaches however result in a correct one-to-one mapping of M(n ? 1) onto S(n). As shown in Table 1, communication between two nodes !; !i 2 V (M) which are adjacent over the ith dimension of M(n ? 1) requires a transposition (r s) which must be properly accomplished by means of star operations that are available in S(n). Let g be the star operation performed along the gth dimension of S(n) (i.e., g exchanges the rst and the gth digit of a permutation of n digits). Assume that transposition (r s) is ordered such that r lt; s. Therefore, (r s) can be minimally executed by the following sequences of star operations:

"... In PAGE 7: ...et the transposition of two digits pr, ps in be denoted by (r s) (i.e., (r s) corresponds to the exchange of the digits occupying the rth and the sth positions in permutation ). Table1 lists the sequences of transpositions used by Algorithm 1 along the (n ? 1) dimensions of the mesh. Note that if the coordinate of the mesh node along dimension i is m[i], then only the rst m[i] transpositions of the sequence corresponding to dimension i are used.... In PAGE 12: ... In the corresponding embedding of M(n ? 1) into S(n), ! and !i+2 are connected by a path containing at most 4 links. Proof : By inspection of Algorithm 1 and Table1 , we note that the mapping of !i+2 2 V (M) onto i+2 2 V (S) uses a sequence of transpositions of the form: i+2 = (a b)(c d) : : :(i j)(k l)(m n)(o p) : : :(y z) Accordingly, ! is mapped onto via a sequence of transpositions of the form: = (a b)(c d) : : :(i j)(o p) : : :(y z) Although identical transpositions are used in and i+2, the mapping sequence of !i+2 has two transpo- sitions more than the mapping sequence of !i. An inspection of Table 1 reveals that the extra transpositions (k l) and (m n) are actually two consecutive transpositions along the ith dimension, whose generic format is (x x + 1)(x ? 1 x) = (s t)(r s); r = s ? 1 = t ? 2.... In PAGE 12: ... Proof : By inspection of Algorithm 1 and Table 1, we note that the mapping of !i+2 2 V (M) onto i+2 2 V (S) uses a sequence of transpositions of the form: i+2 = (a b)(c d) : : :(i j)(k l)(m n)(o p) : : :(y z) Accordingly, ! is mapped onto via a sequence of transpositions of the form: = (a b)(c d) : : :(i j)(o p) : : :(y z) Although identical transpositions are used in and i+2, the mapping sequence of !i+2 has two transpo- sitions more than the mapping sequence of !i. An inspection of Table1 reveals that the extra transpositions (k l) and (m n) are actually two consecutive transpositions along the ith dimension, whose generic format is (x x + 1)(x ? 1 x) = (s t)(r s); r = s ? 1 = t ? 2. According to Equation 6, these two transpositions can be accomplished via a sequence of star operations as follows: (s t)(r s) s ! t !6s !6s = s ! t , if r = 1 s ! t !6s !6s ! r ! s = s ! t ! r ! s , if r 6 = 1 (8) Note that the execution of the two transpositions require either 2 or 4 star operations.... ..."

### Table 1: Mixing model and ICA solution

2006

"... In PAGE 1: ... 2) The use of all the information obtained from the basis vectors solves the permutation problem more accurately and therefore improves the BSS performance. Blind source separation in frequency domain Table1 shows equations related to a mixing model and ICA. Convolutive mixtures in the time domain can be approximated as multiple instantaneous mixtures in the frequency domain.... ..."

Cited by 2

### Table 46: Transitivity bias in each corpus.

"... In PAGE 13: ...able 45: Verbs that WSJ has more of than both Brown and BNC.........................60 Table46 : Transitivity bias in each corpus.... In PAGE 76: ....3.3.2 Results: Nine of the 64 verbs, shown in Table46 , had a significant shift in transitivity bias. These verbs had a different high/mixed/low transitivity bias in at least one of the three corpora.... ..."