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A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

%I A092582 #72 Apr 28 2023 20:08:08
%S A092582 1,1,1,3,2,1,12,8,3,1,60,40,15,4,1,360,240,90,24,5,1,2520,1680,630,
%T A092582 168,35,6,1,20160,13440,5040,1344,280,48,7,1,181440,120960,45360,
%U A092582 12096,2520,432,63,8,1,1814400,1209600,453600,120960,25200,4320,630,80,9,1
%N A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.
%C A092582 Row sums are the factorial numbers (A000142). First column is A001710.
%C A092582 T(n,k) = number of permutations of [n] in which 1,2,...,k is a subsequence but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] in which 12 is a subsequence but 123 is not. - _Emeric Deutsch_, Nov 12 2004
%C A092582 T(n,k) is the number of deco polyominoes of height n with k cells in the last column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column). - _Emeric Deutsch_, Jan 06 2005
%C A092582 T(n,k) is the number of permutations p of [n] for which the smallest i such that p(i)<p(i+1) is k (it is assumed that p(n+1)=infinity). Example: T(4,3)=3 because we have 4312, 4213 and 3214. - _Emeric Deutsch_, Feb 23 2008
%C A092582 Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). - _Emeric Deutsch_, Feb 23 2008
%C A092582 Matrix inverse of A128227*A154990. - _Mats Granvik_, Feb 08 2009
%C A092582 Differences in the columns of A173333 which counts the n-permutations with an initial ascending run of length at least k. - _Geoffrey Critzer_, Jun 18 2017
%C A092582 The triangle with each row reversed is A130477. - _Michael Somos_, Jun 25 2017
%D A092582 M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.
%H A092582 Alois P. Heinz, <a href="/A092582/b092582.txt">Rows n = 1..141, flattened</a>
%H A092582 E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%H A092582 Olivier Bodini, Antoine Genitrini, and Mehdi Naima, <a href="https://arxiv.org/abs/1808.08376">Ranked Schröder Trees</a>, arXiv:1808.08376 [cs.DS], 2018.
%H A092582 Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
%H A092582 Colin Defant and James Propp, <a href="https://arxiv.org/abs/2002.07144">Quantifying Noninvertibility in Discrete Dynamical Systems</a>, arXiv:2002.07144 [math.CO], 2020.
%H A092582 Emeric 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.
%F A092582 T(n, k) = n!*k/(k+1)! for k<n; T(n, n)=1.
%F A092582 Inverse of:
%F A092582    1;
%F A092582   -1,  1;
%F A092582   -1, -2,  1;
%F A092582   -1, -2, -3,  1;
%F A092582   -1, -2, -3, -4,  1;
%F A092582   ... where A002260 = (1; 1,2; 1,2,3; ...). - _Gary W. Adamson_, Feb 22 2012
%F A092582 T(2n,n) = A092956(n-1) for n>0. - _Alois P. Heinz_, Jun 19 2017
%F A092582 From _Alois P. Heinz_, Dec 17 2021: (Start)
%F A092582 Sum_{k=1..n} k * T(n,k) = A002627(n).
%F A092582 |Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)
%F A092582 From _G. C. Greubel_, Sep 06 2022: (Start)
%F A092582 T(n, 1) = A001710(n).
%F A092582 T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.
%F A092582 T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.
%F A092582 T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.
%F A092582 T(n, n-1) = A000027(n-1).
%F A092582 T(n, n-2) = A005563(n-1), n >= 3. (End)
%F A092582 Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - _Alois P. Heinz_, Apr 28 2023
%e A092582 T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
%e A092582      1;
%e A092582      1,    1;
%e A092582      3,    2,   1;
%e A092582     12,    8,   3,   1;
%e A092582     60,   40,  15,   4,  1;
%e A092582    360,  240,  90,  24,  5,  1;
%e A092582   2520, 1680, 630, 168, 35,  6,  1;
%e A092582   ...
%t A092582 Drop[Drop[Abs[Map[Select[#, # < 0 &] &, Map[Differences, nn = 10; Range[0, nn]! CoefficientList[Series[(Exp[y x] - 1)/(1 - x), {x, 0, nn}], {x, y}]]]], 1], -1] // Grid (* _Geoffrey Critzer_, Jun 18 2017 *)
%o A092582 (PARI) {T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* _Michael Somos_, Jun 25 2017 */
%o A092582 (GAP) Flat(List([1..11],n->Concatenation([1],List([1..n-1],k->Factorial(n)*k/Factorial(k+1))))); # _Muniru A Asiru_, Jun 10 2018
%o A092582 (Magma)
%o A092582 A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
%o A092582 [A092582(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 06 2022
%o A092582 (SageMath)
%o A092582 def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
%o A092582 flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Sep 06 2022
%Y A092582 Cf. A002260. - _Gary W. Adamson_, Feb 22 2012
%Y A092582 Cf. A000166, A002627, A055596, A092580, A092956, A130477.
%Y A092582 Cf. A000027, A001710, A001715, A001720, A001725, A005563.
%Y A092582 Cf. A000522.
%K A092582 nonn,tabl
%O A092582 1,4
%A A092582 _Emeric Deutsch_ and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004