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%I #28 Sep 08 2022 08:46:03
%S 0,1,1,10,15,5,90,165,90,15,840,1750,1225,350,35,8400,19180,15750,
%T 5950,1050,70,90720,222264,204624,92610,22050,2646,126,1058400,
%U 2744280,2757720,1421490,411600,67620,5880,210,13305600,36162720,38980920,22203720,7408170,1496880,180180,11880,330
%N Triangle read by rows: T(n,k) is the number of stretching pairs in all permutations in S_{n,k} (=set of permutations in S_n with k cycles) (n >= 3; 1 <= k <= n-2).
%C A stretching pair of a permutation p in S_n is a pair (i,j) (1 <= i < j <= n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because 1< p(2) < 2 < 4 < p(4) = 5.
%C Number of entries in row n (n >= 3) is n - 2.
%C Sum of entries in row n is A216119(n).
%C T(n,1) = A061206(n-3).
%D E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)
%H Muniru A Asiru, <a href="/A216118/b216118.txt">Rows n=3..100 of triangle, flattened</a>
%H E. Clark and R. Ehrenborg, <a href="http://dx.doi.org/10.1016/j.ejc.2008.11.014">Explicit expressions for the extremal excedance statistic</a>, European J. Combinatorics, 31, 2010, 270-279.
%H J. Cooper, E. Lundberg, and B. Nagle, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p28">Generalized pattern frequency in large permutations</a>, Electron. J. Combin. 20, 2013, #P28.
%F T(n,k) = binomial(n,4)*abs(Stirling1(n-2,k)).
%F T(n,k) = binomial(n,4)*(-1)^(n-k)*Stirling1(n-2,k).
%e T(4,1) = 1, T(4,2) = 1 because 22 permutations in S_4 have no stretching pairs, the 1-cycle 3142 has the stretching pair (2,3) and the 2-cycle 2143 has the stretching pair (2,3).
%e Triangle starts:
%e 0;
%e 1, 1;
%e 10, 15, 5;
%e 90, 165, 90, 15;
%e 840, 1750, 1225, 350, 35;
%e ...
%p with(combinat): T := proc (n, k) options operator, arrow: binomial(n, 4)*abs(stirling1(n-2, k)) end proc: for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do; # yields sequence in triangular form
%t T[n_, k_] := Binomial[n, 4] * Abs[StirlingS1[n-2, k]]; Table[T[n, k], {n, 3, 12}, {k, 1, n-2}] // Flatten (* _Amiram Eldar_, Dec 13 2018 *)
%o (GAP) List([3..12],n->List([1..n-2],k->Binomial(n,4)*Stirling1(n-2,k))); # _Muniru A Asiru_, Dec 13 2018
%o (PARI) {T(n,k) = (-1)^(n-k)*binomial(n,4)*stirling(n-2,k,1)};
%o for(n=3, 10, for(k=1,n-2, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 13 2018
%o (Magma) [[(-1)^(n-k)*Binomial(n,4)*StirlingFirst(n-2,k): k in [1..n-2]]: n in [3..12]]; // _G. C. Greubel_, Dec 13 2018
%o (Sage) [[binomial(n,4)*stirling_number1(n-2,k) for k in (1..n-2)] for n in (3..12)] # _G. C. Greubel_, Dec 13 2018
%Y Cf. A216119, A061206.
%K nonn,tabl
%O 3,4
%A _Emeric Deutsch_, Feb 26 2013
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