From zeilberg Thu Jul 18 11:45:12 1996 Received: (from zeilberg@localhost) by euclid.math.temple.edu (8.6.12/8.6.12) id LAA04380 for tewodros; Thu, 18 Jul 1996 11:45:12 -0400 Date: Thu, 18 Jul 1996 11:45:12 -0400 From: Doron Zeilberger Posted-Date: Thu, 18 Jul 1996 11:45:12 -0400 Received-Date: Thu, 18 Jul 1996 11:45:12 -0400 Message-Id: <199607181545.LAA04380@euclid.math.temple.edu> To: tewodros Subject: Propp's problem Content-Length: 2663 X-Lines: 75 Status: RO >From propp@math.mit.edu Mon Jul 1 15:48:11 1996 Received: from math.mit.edu (MATH.MIT.EDU [18.87.0.8]) by euclid.math.temple.edu (8.6.12/8.6.12) with ESMTP id PAA05818 for ; Mon, 1 Jul 1996 15:48:09 -0400 Posted-Date: Mon, 1 Jul 1996 15:48:09 -0400 Received-Date: Mon, 1 Jul 1996 15:48:09 -0400 Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.7.3/8.7.3) with ESMTP id PAA00335 for ; Mon, 1 Jul 1996 15:47:56 -0400 (EDT) Received: (from propp@localhost) by pfaff.mit.edu (8.7.3/8.6.9) id PAA20350 for zeilberg@euclid.math.temple.edu; Mon, 1 Jul 1996 15:47:56 -0400 (EDT) Date: Mon, 1 Jul 1996 15:47:56 -0400 (EDT) From: Jim Propp Message-Id: <199607011947.PAA20350@pfaff.mit.edu> To: zeilberg@euclid.math.temple.edu Subject: determinant Status: RO Doron, >Dear Jim, I enjoyed your talk. Thanks. >Could you please send me >any determinants that evaluate nicely that you can't do >yet? Well, the ones that arise in the tilings problem are huge matrices of 1's, -1's, and 0's whose entries I have no explicit recipe for (though writing down such recipes wouldn't be hard); I rely on the computer to turn pictures into adjacency matrices M and then to change the signs of strategically chosen entries so that the permanent of M equals the determinant of the new matrix K. But here is a determinant that did grow out of some work Greg and I did on tilings back in 1990: It appears that the determinant of the (n+1)-by-(n+1) matrix (indexed by i,j running from 0 to n) whose (i,j)th entry is (i+j)! (2n-i-j)! ------------------- i! j! (n-i)! (n-j)! is equal to n n-1 n-2 1 (2n+1) (2n) (2n-1) ... (n+2) -------------------------------------- . 1 2 3 n (n) (n-1) (n-2) ... (1) This formula works up to n=9. More generally, if one lets M denote the (m+1)-by-(n+1) matrix whose (i,j)th entry is (i+j)! (m+n-i-j)! ------------------- i! j! (m-i)! (n-j)! then the determinant of M-transpose times M appears to be 2n+2 [(m+n+1)!] H(m-n) --------------------- , H(2n+2) H(m+n+2) where H(n) = 1!2!...(n-1)!. Greg has shown that the formula has a nice q-analogue, though he hasn't nailed down the exact form of the right hand side, as far as I know. I should mention that we do not need this conjecture for anything. In fact, these matrices have no combinatorial interpretation that I know of --- though they resemble certain matrices that did arise from some applications (which at this point I have forgotten). Jim