The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings. 15
 1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums give A000142. First column gives A000108. Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4). Also the number of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2). - Alois P. Heinz, Nov 14 2015 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. S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163. FindStat - Combinatorial Statistic Finder, The number of nestings of a permutation, The number of crossings of a permutation R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2. Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021. FORMULA Sum_{k>0} k * T(n,k) = A001754(n). T(n,n) = A287328(n). - Alois P. Heinz, Aug 31 2017 EXAMPLE Triangle begins: 0 : 1; 1 : 1; 2 : 2; 3 : 5, 1; 4 : 14, 8, 2; 5 : 42, 45, 25, 7, 1; 6 : 132, 220, 198, 112, 44, 12, 2; 7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1; ... 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^(j-1)), 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); # Alois P. Heinz, Nov 14 2015 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^(j-1)], {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 31 2016, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497. Cf. A000142, A001754, A002620, A260665, A260670, A287328, A291722. Sequence in context: A319120 A274404 A101282 * A145879 A231210 A178978 Adjacent sequences: A263773 A263774 A263775 * A263777 A263778 A263779 KEYWORD nonn,tabf AUTHOR Christian Stump, Oct 26 2015 EXTENSIONS More terms from Alois P. Heinz, Oct 26 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 23 16:36 EDT 2024. Contains 372765 sequences. (Running on oeis4.)