%I #72 Apr 28 2023 20:08:08

%S 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 168,35,6,1,20160,13440,5040,1344,280,48,7,1,181440,120960,45360,

%U 12096,2520,432,63,8,1,1814400,1209600,453600,120960,25200,4320,630,80,9,1

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

%C Row sums are the factorial numbers (A000142). First column is A001710.

%C 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 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 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 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 Matrix inverse of A128227*A154990. - _Mats Granvik_, Feb 08 2009

%C 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 The triangle with each row reversed is A130477. - _Michael Somos_, Jun 25 2017

%D M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.

%H Alois P. Heinz, <a href="/A092582/b092582.txt">Rows n = 1..141, flattened</a>

%H 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 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 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 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 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 T(n, k) = n!*k/(k+1)! for k<n; T(n, n)=1.

%F Inverse of:

%F 1;

%F -1, 1;

%F -1, -2, 1;

%F -1, -2, -3, 1;

%F -1, -2, -3, -4, 1;

%F ... where A002260 = (1; 1,2; 1,2,3; ...). - _Gary W. Adamson_, Feb 22 2012

%F T(2n,n) = A092956(n-1) for n>0. - _Alois P. Heinz_, Jun 19 2017

%F From _Alois P. Heinz_, Dec 17 2021: (Start)

%F Sum_{k=1..n} k * T(n,k) = A002627(n).

%F |Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)

%F From _G. C. Greubel_, Sep 06 2022: (Start)

%F T(n, 1) = A001710(n).

%F T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.

%F T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.

%F T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.

%F T(n, n-1) = A000027(n-1).

%F T(n, n-2) = A005563(n-1), n >= 3. (End)

%F Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - _Alois P. Heinz_, Apr 28 2023

%e T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 12, 8, 3, 1;

%e 60, 40, 15, 4, 1;

%e 360, 240, 90, 24, 5, 1;

%e 2520, 1680, 630, 168, 35, 6, 1;

%e ...

%t 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 (PARI) {T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* _Michael Somos_, Jun 25 2017 */

%o (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 (Magma)

%o A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;

%o [A092582(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 06 2022

%o (SageMath)

%o def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)

%o flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Sep 06 2022

%Y Cf. A002260. - _Gary W. Adamson_, Feb 22 2012

%Y Cf. A000166, A002627, A055596, A092580, A092956, A130477.

%Y Cf. A000027, A001710, A001715, A001720, A001725, A005563.

%Y Cf. A000522.

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_ and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004