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A216118
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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).
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2
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0, 1, 1, 10, 15, 5, 90, 165, 90, 15, 840, 1750, 1225, 350, 35, 8400, 19180, 15750, 5950, 1050, 70, 90720, 222264, 204624, 92610, 22050, 2646, 126, 1058400, 2744280, 2757720, 1421490, 411600, 67620, 5880, 210, 13305600, 36162720, 38980920, 22203720, 7408170, 1496880, 180180, 11880, 330
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OFFSET
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3,4
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COMMENTS
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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.
Number of entries in row n (n >= 3) is n - 2.
Sum of entries in row n is A216119(n).
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REFERENCES
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E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)
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LINKS
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FORMULA
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T(n,k) = binomial(n,4)*abs(Stirling1(n-2,k)).
T(n,k) = binomial(n,4)*(-1)^(n-k)*Stirling1(n-2,k).
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EXAMPLE
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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).
Triangle starts:
0;
1, 1;
10, 15, 5;
90, 165, 90, 15;
840, 1750, 1225, 350, 35;
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(GAP) List([3..12], n->List([1..n-2], k->Binomial(n, 4)*Stirling1(n-2, k))); # Muniru A Asiru, Dec 13 2018
(PARI) {T(n, k) = (-1)^(n-k)*binomial(n, 4)*stirling(n-2, k, 1)};
for(n=3, 10, for(k=1, n-2, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 13 2018
(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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