Search: id:A092582
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%I A092582
%S A092582 1,1,1,3,2,1,12,8,3,1,60,40,15,4,1,360,240,90,24,5,1,2520,1680,630,168,
%T A092582 35,6,1,20160,13440,5040,1344,280,48,7,1,181440,120960,45360,12096,2520,
%U A092582 432,63,8,1,1814400,1209600,453600,120960,25200,4320,630,80,9,1
%N A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] 
               having length of first run equal to k.
%C A092582 Row sums are the factorial numbers (A000142). First column is A001710.
%C A092582 T(n,k)=number of permutations of [n] in which 1,2,...,k is a subsequence 
               but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 
               4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] 
               in which 12 is a subsequence but 123 is not. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 12 2004
%C A092582 T(n,k) is the number of deco polyominoes of height n with k cells in 
               the last column. (A deco polyomino is a directed column-convex polyomino 
               in which the height, measured along the diagonal, is attained only 
               in the last column). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jan 06 2005
%C A092582 T(n,k) is the number of permutations p of [n] for which the smallest 
               i such that p(i)<p(i+1) is k (it is assumed that p(n+1)=infinity). 
               Example: T(4,3)=3 because we have 4312, 4213 and 3214. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Feb 23 2008
%C A092582 Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,
               44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2008
%C A092582 Matrix inverse of A128227*A154990. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Feb 08 2009]
%D A092582 E. Barcucci, A. del Lungo and R. Pinzani, "Deco" polyominoes, permutations 
               and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%D A092582 E. Deutsch and W. P. Johnson, Create your own permutation statistic, 
               Math. Mag., 77, 130-134, 2004.
%D A092582 M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, 
               Florida, 2004.
%F A092582 T(n, k)=n!*k/(k+1)! for k<n; T(n, n)=1.
%e A092582 T(4,3)=3 because 1243, 1342 and 2341 are the only permutations of [4] 
               having length of first run equal to 3.
%e A092582 1; 1,1; 3,2,1; 12,8,3,1; 60,40,15,4,1; 360,240,90,24,5,1; 2520,1680,630,
               168,35,6,1;
%Y A092582 Cf. A000166.
%Y A092582 Sequence in context: A115085 A110616 A059418 this_sequence A068440 A048647 
               A059438
%Y A092582 Adjacent sequences: A092579 A092580 A092581 this_sequence A092583 A092584 
               A092585
%K A092582 nonn,tabl
%O A092582 1,4
%A A092582 Emeric Deutsch (deutsch(AT)duke.poly.edu) and Warren P. Johnson (wjohnson(AT)bates.edu), 
               Apr 10 2004

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