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 A260670 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows. 13
 1, 1, 2, 5, 1, 15, 6, 3, 52, 32, 23, 10, 3, 203, 171, 152, 98, 62, 22, 11, 1, 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4, 4140, 5444, 6460, 6082, 5513, 4302, 3328, 2141, 1393, 780, 432, 187, 88, 24, 6, 21147, 32919, 43626, 46508, 46880, 41979, 36774 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Patterns 1-32, 3-12, 21-3 give the same sequence. LINKS Alois P. Heinz, Rows n = 0..50, flattened A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001 FORMULA Sum_{k>0} k * T(n,k) = A001754(n). EXAMPLE T(3,1) = 1: 231. T(4,1) = 6: 1342, 2314, 2413, 2431, 3241, 4231. T(4,2) = 3: 2341, 3412, 3421. T(5,2) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421. T(5,3) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321. T(5,4) = 3: 34512, 34521, 45231. Triangle T(n,k) begins: 0 : 1; 1 : 1; 2 : 2; 3 : 5, 1; 4 : 15, 6, 3; 5 : 52, 32, 23, 10, 3; 6 : 203, 171, 152, 98, 62, 22, 11, 1; 7 : 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4; MAPLE b:= proc(u, o) option remember; `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+ add(expand(b(u+j-1, o-j)*x^u), j=1..o)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)): seq(T(n), n=0..10); MATHEMATICA b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^u], {j, 1, o}]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 16 2017, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000110, A264460, A264461, A264462, A264463, A264464, A264465, A264466, A264467, A264468, A264469. Row sums give A000142. Cf. A001754, A125811, A260665, A263776. Sequence in context: A231210 A178978 A101895 * A260665 A110220 A349106 Adjacent sequences: A260667 A260668 A260669 * A260671 A260672 A260673 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Nov 14 2015 STATUS approved

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Last modified June 23 01:26 EDT 2024. Contains 373629 sequences. (Running on oeis4.)