OFFSET
1,4
COMMENTS
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
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
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
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
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
The triangle with each row reversed is A130477. - Michael Somos, Jun 25 2017
REFERENCES
M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Colin Defant and James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
Emeric Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.
FORMULA
T(n, k) = n!*k/(k+1)! for k<n; T(n, n)=1.
Inverse of:
1;
-1, 1;
-1, -2, 1;
-1, -2, -3, 1;
-1, -2, -3, -4, 1;
... where A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Feb 22 2012
T(2n,n) = A092956(n-1) for n>0. - Alois P. Heinz, Jun 19 2017
From Alois P. Heinz, Dec 17 2021: (Start)
Sum_{k=1..n} k * T(n,k) = A002627(n).
|Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)
From G. C. Greubel, Sep 06 2022: (Start)
T(n, 1) = A001710(n).
T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.
T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.
T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.
T(n, n-1) = A000027(n-1).
T(n, n-2) = A005563(n-1), n >= 3. (End)
EXAMPLE
T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
1;
1, 1;
3, 2, 1;
12, 8, 3, 1;
60, 40, 15, 4, 1;
360, 240, 90, 24, 5, 1;
2520, 1680, 630, 168, 35, 6, 1;
...
MATHEMATICA
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 *)
PROG
(PARI) {T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* Michael Somos, Jun 25 2017 */
(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
(Magma)
A092582:= func< n, k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
[A092582(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 06 2022
(SageMath)
def A092582(n, k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
flatten([[A092582(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Sep 06 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004
STATUS
editing