C. Krattenthaler, Advanced determinant calculus, Semin. Lothar. Comb. 42 (1999), B42q.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the case = 2, we obviously have k = 1 ( 2 1) 1 (2 2 1 1) Setting = m h 1; m) we retrieve the formula (7) of Chen [1] The determinant in (12) is, in general, not easy to evaluate. For example, none of the recipes in the rather comprehensive compendium of Krattenthaler [12] seems applicable. However, in a special case, a much simpler formula can be obtained. Theorem 7. Suppose 1 ; are in an arithmetic progression, i.e. there are c; d 2 Z such that i = c(i 1) d for i = 1; Then ; where 1 : c d = 2 (1 ) 1 : The key ....

C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin. 42 (1999), Article B42q, 67 pp.

....of the appropriate orthogonal polynomials allows one to evaluate the determinant (k = 1) and sometimes the 4 fold hyperdeterminant. 3. Examples derived from Selberg s integral The evaluation of Hankel determinants built on classical sequences of combinatorial numbers arises in many contexts (see [20, 40] and references therein) Recent work on the theory of coherent states has led to the discovery of integral representations of many such sequences [31, 32] For the sequences proportional to moments of a Beta distribution, the Hankel hyperdeterminants can be evaluated immediately. We have B(a n; ....

C Krattenthaler, Advanced determinant calculus, in The Andrews Festschrift, Seminaire Lotharingien de Combinatoire 42 (1998), electronic, 67 pp.

....would identify that square numbers: and triangle numbers: had been composed with repetition to form this sequence. Hofstadter aimed to model how humans search for reasonable definitions of sequences, rather than to provide a tool to identify sequences. The Guess program (Krattenthaler 1991) is such a tool which uses techniques from determinant calculus to produce a closed form definition for a given sequence. We have recently applied HR to the problem of identification and extrapolation of integer sequences (Colton, Bundy, Walsh 2000) which requires more search control and is a ....

Krattenthaler, C. 1991. Advanced determinant calculus. Technical report, Institute of Mathematics, University of Vienna.

....would identify that square numbers: 1; 4; 9; and triangle numbers: 1; 3; 6; had been composed with repetition to form this sequence. Hofstadter aimed to model how humans search for reasonable definitions of sequences, rather than to provide a tool to identify sequences. The Guess program (Krattenthaler 1991) is such a tool which uses techniques from determinant calculus to produce a closed form definition for a given sequence. We have recently applied HR to the problem of identification and extrapolation of integer sequences (Colton, Bundy, Walsh 2000) which requires more search control and is a ....

Krattenthaler, C. 1991. Advanced determinant calculus. Technical report, Institute of Mathematics, University of Vienna.

....entries, see Lemma 1. In case that the xed rhombus is situated in the centre or next to the centre we are able to evaluate the determinant, see Lemma 3 and Lemma 4 in Section 4. We partly make us of a method that has already produced evaluations of other binomial determinants (see e.g. 4] [9]) In the course of evaluating the determinant we need an alternative expression for the number of rhombus tilings that we are interested in, in form of the aforementioned triple sum. This is given in Lemma 2 in Section 3. Finally, in Section 5 we provide the proofs of Theorem 3 and Theorem 4. 2. ....

C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (1999), to appear.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 67 pp.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), Article B42q, 67 pp.

....from the Ansatz (4.1) 4.3) are related to the coecients of P (z) resulting from the Ansatz (3.4) Since this is not essential for the proof of Theorem 1, we leave the details to the reader. Proof of Theorem 6. We follow the identi cation of factors method as described in Section 2. 4 in [5]. First we show that (2X 1;a (32k 2a 1)Y a ) divides det M , a = 1; 2; 4k 1. What has to be proved is that det M vanishes for X 1;a = 32k 2a 1)Y a =2. This can be done by showing that for this choice of X 1;a there is a nontrivial linear combination of the rows of M . ....

C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), Article B42q, 67 pp.

....de nition, the vector of coecients of Q(z) Using Chu Vandermonde summation (see e.g. 22, Sec. 5.1, 5.27) again, the numerator on the righthand side of (3) specialized at z = 1 is n 2r i j n 2r i j 1 : 12) This determinant can be evaluated by using e.g. Theorem 30, 3. 18) in [32], with n replaced by r, q = 1, A = 2n 4r, and L i = n 2r i. The result is (2n 4r 2i 2) n r i) n r i 1) 1 i j r (j i) 1 i j r (i j 1) This expression can be transformed into the one given in (11) ut Example. From the Structure Theorem for Gorenstein ideals of ....

....is the n th Catalan number 2n . Desainte Catherine and Viennot evaluate this determinant by means of the quotient di erence algorithm. However, there are much easier ways to do it, for example, by using the fact that Cn = 1) 2n 1 1=2 and noticing that therefore Theorem 26, 3. 12) in [32] is applicable. This latter observation shows that actually a more general determinant, det 1 i;j r (C i j ) can be evaluated. This determinant appears also in connection with tableaux counting, see [18, second half of Sec. 9] A weighted version of the tableaux counting problem of ....

C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 67 pp.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 66 pp.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 66 pp.

....from the Ansatz (4.1) 4.3) are related to the coecients of P (t) resulting from the Ansatz (3.4) Since this is not essential for the proof of Theorem 1, we leave the details to the reader. Proof of Theorem 6. We follow the identi cation of factors method as described in Section 2. 4 in [5]. First we show that (2X 1;a (32k 2 2a 1)Y a ) divides det M X , a = 1; 2; 4k 1. What has to be proved is that det M X vanishes for X 1;a = 32k 2 2a 1)Y a =2. This can be done by showing that for this choice of X 1;a there is a nontrivial linear combination of the rows of M ....

C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), Article B42q, 67 pp.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 67 pp.

....of Q(z) Using Chu Vandermonde summation (see e.g. 22, Sec. 5.1, 5.27) again, the numerator on the righthand side of (3) specialized at z = 1 is det 1 i;j r 2n 4r n 2r i j 2n 4r n 2r i j 1 : 12) This determinant can be evaluated by using e.g. Theorem 30, 3. 18) in [31], with n replaced by r, q = 1, A = 2n 4r, and L i = n 2r i. The result is r Y i=1 (2n 4r 2i 2) n r i) n r i 1) Y 1 i j r (j i) Y 1 i j r (i j 1) This expression can be transformed into the one given in (11) ut The Hilbert series of Pfaan rings 13 Example. From the ....

....1 n 1 2n n . Desainte Catherine and Viennot evaluate this determinant by means of the quotient di erence algorithm. However, there are much easier ways to do it, for example, by using the fact that Cn = 1) n 2 2n 1 1=2 n 1 and noticing that therefore Theorem 26, 3. 12) in [31] is applicable. This latter observation shows that actually a more general determinant, det 1 i;j r (C i j ) can be evaluated. This determinant appears also in connection with tableaux counting, see [18, second half of Sec. 9] A weighted version of the tableaux counting problem of ....

C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin. 42 (\The Andrews Festschrift") (1999), paper B42q, 67 pp.

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C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (\The Andrews Festschrift") (1999), Article B42q, 67 pp.

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C. Krattenthaler, Advanced determinant calculus, Semin. Lothar. Comb. 42 (1999), B42q.

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C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin. 42 (1999), B42q.

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Krattenthaler, C., Advanced determinant calculus, Sem. Lothar. Comb. 42 (1999), .

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C. Krattenthaler, Advanced determinant calculus. The Andrews Festschrift (Maratea, 1998). Sem. Lothar. Combin. 42 (1999), Art. B42q, 67 pp. (electronic).

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C. Krattenthaler, Advanced Determinant Calculus, Semin. Lothar. Comb. 42, B42q, 67 pages.

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