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A092583 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.

%I A092583 #21 Sep 08 2022 08:45:13
%S A092583 1,0,2,0,1,5,0,4,6,14,0,20,30,28,42,0,120,180,168,120,132,0,840,1260,
%T A092583 1176,840,495,429,0,6720,10080,9408,6720,3960,2002,1430,0,60480,90720,
%U A092583 84672,60480,35640,18018,8008,4862,0,604800,907200,846720,604800,356400,180180,80080,31824,16796
%N A092583 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.
%C A092583 Row sums are the factorial numbers (A000142).
%C A092583 Diagonal is A000108.
%C A092583 T(n,n-1) = binomial(2n-2,n-3) = A002694(n-1).
%H A092583 G. C. Greubel, <a href="/A092583/b092583.txt">Rows n = 1..100 of triangle, flattened</a>
%H A092583 E. Deutsch and W. P. Johnson, <a href="http://www.jstor.org/stable/3219101">Create your own permutation statistics</a>, Math. Mag., 77, 130-134, 2004.
%H A092583 R. Simion and F. W. Schmidt, <a href="https://doi.org/10.1016/S0195-6698(85)80052-4">Restricted permutations</a>, European J. Combin., 6, 383-406, 1985.
%F A092583 T(n,k) = n!*binomial(2k, k-2)/(k+1)! for k < n;
%F A092583 T(n,n) = binomial(2n, n)/(n+1) = A000108(n).
%e A092583 T(4,3) = 6 because 1324, 1423, 2134, 2314, 3124 and 4123 are the only permutations of [4] in which the length of the longest initial segment avoiding the 123-pattern is equal to 3 (i.e., the first three entries do not contain the 123-pattern but all 4 of them do).
%e A092583 Triangle starts:
%e A092583   1;
%e A092583   0,    2;
%e A092583   0,    1,    5;
%e A092583   0,    4,    6,   14;
%e A092583   0,   20,   30,   28,   42;
%e A092583   0,  120,  180,  168,  120,  132;
%e A092583   0,  840, 1260, 1176,  840,  495,  429;
%e A092583   ...
%t A092583 T[n_,k_]:= If[k==n, CatalanNumber[n], n!*Binomial[2*k,k-2]/(k+1)!]; Table[T[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 22 2019 *)
%o A092583 (PARI) tabl(nn) = {for (n=1, nn, for (k=1, n-1, print1(n!*binomial(2*k, k-2)/(k+1)!, ", ");); print1(binomial(2*n, n)/(n+1), ", "); print(););} \\ _Michel Marcus_, Jul 16 2013
%o A092583 (Magma)
%o A092583 T:= func< n,k | k eq n select Catalan(n) else Factorial(n)*Binomial(2*k, k-2)/Factorial(k+1) >;
%o A092583 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 22 2019
%o A092583 (Sage)
%o A092583 def T(n, k):
%o A092583     if (k==n): return catalan_number(n)
%o A092583     else: return factorial(n)*binomial(2*k, k-2)/factorial(k+1)
%o A092583 [[T(n,k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 22 2019
%o A092583 (GAP)
%o A092583 T:= function(n,k)
%o A092583     if k=n then return Binomial(2*n, n)/(n + 1);
%o A092583     else return Factorial(n)*Binomial(2*k, k-2)/Factorial(k+1);
%o A092583     fi;
%o A092583   end;
%o A092583 Flat(List([1..12], n-> List([1..n], k-> T(n,k) ))); # _G. C. Greubel_, Jul 22 2019
%Y A092583 Cf. A000142, A000108, A002694.
%K A092583 nonn,tabl
%O A092583 1,3
%A A092583 _Emeric Deutsch_ and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004