%I A122843
%S A122843 1,2,1,7,4,1,32,21,6,1,180,130,41,8,1,1200,930,312,67,10,1,9240,7560,
%T A122843 2646,602,99,12,1,80640,68880,24864,5880,1024,137,14,1,786240,695520,
%U A122843 257040,62496,11304,1602,181,16,1,8467200,7711200,2903040,720720,133920
%N A122843 Triangle read by rows: T[n,k] = the number of ascending runs of length 
               k in the permutations of [n] for k <= n.
%C A122843 Also T[n,k] = number of rising sequences of length k among all permutations. 
               E.g. T[4,3]=6 because in the 24 permutations of n=4, there are 6 
               rising sequences of length 3: {1,2,3} in {1,2,4,3}, {1,2,3} in {1,
               4,2,3}, {2,3,4} in {2,1,3,4}, {2,3,4} in {2,3,1,4}, {2,3,4} in {2,
               3,4,1}, {1,2,3} in {4,1,2,3}. - Harlan J. Brothers (harlan(AT)brotherstechnology.com), 
               Jul 23 2008
%C A122843 Further comments and formulae from Harlan J. Brothers (harlan(AT)brotherstechnology.com), 
               Jul 23 2008 (Start): The nth row sums to (n+1)!/2, consistent with 
               total count implied by the nth row in the table of Eulerians, A008292.
%C A122843 Generating this triangle through use of the diagonal polynomials allows 
               one to produce an arbitrary number of "imaginary" columns corresponding 
               to runs of length 0, -1, -2, etc. These columns match A001286, A001048 
               and the factorial function respectively.
%C A122843 As n->inf, there is a limiting value for the count of each length expressed 
               as a fraction of all rising sequences in the permutations of n. The 
               numerators of the set of limit fractions are given by A028387 and 
               the denominators by A001710.
%C A122843 As a table of diagonals d[i]:
%C A122843 d[1][n]=1
%C A122843 d[2][n]=2n
%C A122843 d[3][n]=3n^2+5n-1
%C A122843 d[4][n]=4n^3+18n^2+16n-6
%C A122843 d[5][n]=5n^4+42n^3+106n^2+63n-36
%C A122843 d[6][n]=6n^5+80n^4+374n^3+688n^2+292n-240
%C A122843 T[n,k]= n!(n(k^2+k-1)-k(k^2-4)+1)/(k+2)!+Floor[k/n](1/(k(k+3)+2)), 0<k<=n. 
               E.f.g. for column n: (x^(n+1)((n^2+3n+1)x-2(n^2+2n)))/((n+2)!(x-1)^2) 
               (End)
%D A122843 Persi Diaconis, Mathematical developments from the analysis of riffle 
               shuffling, http://www-stat.stanford.edu/~cgates/PERSI/papers/Riffle.pdf, 
               p.4.
%D A122843 C. M. Grinstead and J. L. Snell, Introduction to Probability, American 
               Mathematical Society, 1997, pp.120-131.
%H A122843 Francis Edward Su, <a href="http://www.math.hmc.edu/funfacts/ffiles/20001.4-6.shtml">
               Rising Sequences in Card Shuffling</a>
%F A122843 T[n,k] = n![(n(k(k+1)-1) - k(k-2)(k+2) + 1]/(k+2)! for 0<k<n; T[n,n] 
               = 1; T[n,k] = A122844(n,k) - A122844(n,k+1)
%e A122843 Triangle begins:
%e A122843 1
%e A122843 2 1
%e A122843 7 4 1 (there are 4 ascending runs of length 2 in the permutations of 
               [3], namely 13 in 132 and in 213, 23 in 231, 12 in 312. T[3,2] = 
               4)
%e A122843 32,21,6,1,
%e A122843 180,130,41,8,1
%e A122843 ...
%t A122843 Table[n!((n(k(k+1)-1)-k(k-2)(k+2)+1))/(k+2)!+Floor[k/n]1/(k(k+3)+2),{n,
               1,10},{k,1,n}]//TableForm - Harlan J. Brothers (harlan(AT)brotherstechnology.com), 
               Jul 23 2008
%Y A122843 Cf. A008292, A097900, A001286, A001048, A000142, A028387, A001710.
%Y A122843 Cf. A122844, A001710, A006157, A005460.
%Y A122843 Sequence in context: A072248 A092276 A011274 this_sequence A167196 A107865 
               A089225
%Y A122843 Adjacent sequences: A122840 A122841 A122842 this_sequence A122844 A122845 
               A122846
%K A122843 easy,nonn,tabl
%O A122843 1,2
%A A122843 David J. Scambler (dscambler(AT)bmm.com), Sep 13 2006