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.
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
Offset: 0
Examples
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; ...
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.
Crossrefs
Programs
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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 *)
Formula
Sum_{k>0} k * T(n,k) = A001754(n).
T(n,n) = A287328(n). - Alois P. Heinz, Aug 31 2017
Extensions
More terms from Alois P. Heinz, Oct 26 2015
Comments