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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.

%I A263776 #60 Jul 16 2021 08:35:27
%S A263776 1,1,2,5,1,14,8,2,42,45,25,7,1,132,220,198,112,44,12,2,429,1001,1274,
%T A263776 1092,700,352,140,42,9,1,1430,4368,7280,8400,7460,5392,3262,1664,716,
%U A263776 256,74,16,2,4862,18564,38556,56100,63648,59670,47802,33338,20466,11115
%N 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.
%C A263776 Row sums give A000142.
%C A263776 First column gives A000108.
%C A263776 Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4).
%C A263776 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
%H A263776 Alois P. Heinz, <a href="/A263776/b263776.txt">Rows n = 0..50, flattened</a>
%H A263776 A. Claesson and T. Mansour, <a href="http://arxiv.org/abs/math/0110036">Counting occurrences of a pattern of type (1,2) or (2,1) in permutations</a>, arXiv:math/0110036 [math.CO], 2001.
%H A263776 S. Corteel, <a href="http://dx.doi.org/10.1016/j.aam.2006.01.006">Crossings and alignments of permutations</a>, Adv. Appl. Math 38 (2007) 149-163.
%H A263776 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000223">The number of nestings of a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000039">The number of crossings of a permutation</a>
%H A263776 R. Parviainen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Parviainen/parviainen3.html">Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
%H A263776 Lucas Sá and Antonio M. García-García, <a href="https://arxiv.org/abs/2104.07647">The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos</a>, arXiv:2104.07647 [hep-th], 2021.
%F A263776 Sum_{k>0} k * T(n,k) = A001754(n).
%F A263776 T(n,n) = A287328(n). - _Alois P. Heinz_, Aug 31 2017
%e A263776 Triangle begins:
%e A263776 0 :   1;
%e A263776 1 :   1;
%e A263776 2 :   2;
%e A263776 3 :   5,    1;
%e A263776 4 :  14,    8,    2;
%e A263776 5 :  42,   45,   25,    7,   1;
%e A263776 6 : 132,  220,  198,  112,  44,  12,   2;
%e A263776 7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1;
%e A263776 ...
%p A263776 b:= proc(u, o) option remember;
%p A263776       `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
%p A263776        add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
%p A263776     end:
%p A263776 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A263776 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 14 2015
%t A263776 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_ *)
%Y A263776 Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497.
%Y A263776 Cf. A000142, A001754, A002620, A260665, A260670, A287328, A291722.
%K A263776 nonn,tabf
%O A263776 0,3
%A A263776 _Christian Stump_, Oct 26 2015
%E A263776 More terms from _Alois P. Heinz_, Oct 26 2015