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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

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Last modified August 14 10:58 EDT 2022. Contains 356116 sequences. (Running on oeis4.)