Search: id:A001100
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%I A001100
%S A001100 1,0,2,0,4,2,2,10,10,2,14,40,48,16,2,90,230,256,120,22,2,646,1580,1670,
%T A001100 888,226,28,2,5242,12434,12846,7198,2198,366,34,2,47622,110320,112820,
               64968,
%U A001100 22120,4448,540,40,2,479306,1090270,1108612,650644,236968,54304,7900,748
%N A001100 Triangle read by rows: T(n,k) = number of permutations of length n with 
               exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.
%C A001100 Number of permutations of 12...n such that exactly k of the following 
               occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
%D A001100 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 263.
%D A001100 J. Riordan, A recurrence for permutations without rising or falling successions. 
               Ann. Math. Statist. 36 (1965), 708-710.
%D A001100 David Sankoff and Lani Haque, Power Boosts for Cluster Tests, in Comparative 
               Genomics, Lecture Notes in Computer Science, Volume 3678/2005, Springer-Verlag. 
               [Added by N. J. A. Sloane, Jul 09 2009]
%F A001100 Let T{n, k} = number of permutations of 12...n with exactly k rising 
               or falling successions. Let S[n](t) = Sum_{k >= 0} T{n, k}*t^k. Then 
               S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] 
               = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] 
               + (1-t)^3*(n-3)*S[n-4].
%e A001100 1; 0,2; 0,4,2; 2,10,10,2; 14,40,48,16,2; ...
%Y A001100 Diagonals give A002464, A086852, A086853, A086854, A086955.
%Y A001100 Triangle in A086856 multiplied by 2. Cf. A010028.
%Y A001100 Sequence in context: A037035 A159984 A112824 this_sequence A136265 A066910 
               A094405
%Y A001100 Adjacent sequences: A001097 A001098 A001099 this_sequence A001101 A001102 
               A001103
%K A001100 tabl,nonn
%O A001100 1,3
%A A001100 N. J. A. Sloane (njas(AT)research.att.com), Aug 19 2003

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