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.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
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
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.
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
Sequence in context: A319120 A274404 A101282 * A145879 A231210 A178978
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.

License Agreements, Terms of Use, Privacy Policy. .

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