A062869 Triangle read by rows: For n >= 0, k >= 0, T(n,k) is the number of permutations pi of n such that the total distance Sum_i abs(i-pi(i)) = 2k. Equivalently, k = Sum_i max(i-pi(i),0).

1, 1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 4, 1, 4, 12, 24, 35, 24, 20, 1, 5, 18, 46, 93, 137, 148, 136, 100, 36, 1, 6, 25, 76, 187, 366, 591, 744, 884, 832, 716, 360, 252, 1, 7, 33, 115, 327, 765, 1523, 2553, 3696, 4852, 5708, 5892, 5452, 4212, 2844, 1764, 576, 1, 8, 42, 164, 524

Offset

0,6

Comments

Number of possible values is 1,2,3,5,7,10,13,17,21,... = A033638. Maximum distance divided by 2 is the same minus one, i.e., 0,1,2,4,6,9,12,16,20,... = A002620.

T. Kyle Petersen and Bridget Eileen Tenner proved that T(n,k) is also the number of permutations of n for which the sum of descent differences equals k. - Susanne Wienand, Sep 11 2014

References

D. E. Knuth, The Art of Computer Programming, vol. 3, (1998), page 22 (exercise 28) and page 597 (solution and comments).

Links

Daniel Graf and Alois P. Heinz, Rows n = 0..50, flattened (first 31 rows from Alois P. Heinz)

Max Alekseyev, Proof that T(n,k) is even for k>=n, SeqFan Mailing List, Dec 07 2006

A. Bärtschi, B. Geissmann, D. Graf, T. Hruz, P. Penna, T. Tschager On computing the total displacement number via weighted Motzkin paths, arXiv:1606.05538 [cs.DS], 2016. - Daniel Graf, Jun 20 2016

P. Diaconis and R. L. Graham, Spearman's Footrule as a Measure of Disarray, J. Royal Stat. Soc. B, 39 (1977), 262-268.

FindStat - Combinatorial Statistic Finder, The depth of a permutation., The sum of the descent differences of a permutations.

Mathieu Guay-Paquet and T. Kyle Petersen, The generating function for total displacement, arXiv:1404.4674 [math.CO], 2014.

M. Guay-Paquet, T. K. Petersen, The generating function for total displacement, The Electronic Journal of Combinatorics, 21(3) (2014), #P3.37.

Dirk Liebhold, G Nebe, A Vazquez-Castro, Network coding and spherical buildings, arXiv preprint arXiv:1612.07177 [cs.IT], 2016.

T. Kyle Petersen and Bridget Eileen Tenner, The depth of a permutation, arXiv:1202.4765 [math.CO], 2012.

Formula

From Mathieu Guay-Paquet, Apr 30 2014: (Start)

G.f.: 1/(1-z/(1-t*z/(1-2*t*z/(1-2*t^2*z/(1-3*t^2*z/(1-3*t^3*z/(1-4*t^3*z/(1-4*t^4*z/(...

This is a continued fraction where the (2i)th numerator is (i+1)*t^i*z, and the (2i+1)st numerator is (i+1)*t^(i+1)*z for i >= 0. The coefficient of z^n gives row n, n >= 1, and the coefficient of t^k gives column k, k >= 0. (End)

From Alois P. Heinz, Oct 02 2022: (Start)

Sum_{k=0..floor(n^2/4)} k * T(n,k) = A005990(n).

Sum_{k=0..floor(n^2/4)} (-1)^k * T(n,k) = A009006(n). (End).

Example

Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7,  9,   4;
  1, 4, 12, 24,  35,  24,  20;
  1, 5, 18, 46,  93, 137, 148, 136, 100,  36;
  1, 6, 25, 76, 187, 366, 591, 744, 884, 832, 716, 360, 252;
  ...
(4,3,1,2) has distances (3,1,2,2), sum is 8 and there are 3 other permutations of degree 4 with this sum.

Maple

# The following program yields the entries of the specified row n
n := 9: with(combinat): P := permute(n): excsum := proc (p) (1/2)*add(abs(p[i]-i), i = 1 .. nops(p)) end proc: f[n] := sort(add(t^excsum(P[j]), j = 1 .. factorial(n))): seq(coeff(f[n], t, j), j = 0 .. floor((1/4)*n^2)); # Emeric Deutsch, Apr 02 2010
# Maple program using the g.f. given by Guay-Paquey and Petersen:
g:= proc(h, n) local i, j; j:= irem(h, 2, 'i');
       1-`if`(h=n, 0, (i+1)*z*t^(i+j)/g(h+1, n))
    end:
T:= n-> (p-> seq(coeff(p, t, k), k=0..degree(p)))
        (coeff(series(1/g(0, n), z, n+1), z, n)):
seq(T(n), n=0..10);  # Alois P. Heinz, May 02 2014

Mathematica

g[h_, n_] := Module[{i, j}, {i, j} = QuotientRemainder[h, 2]; 1 - If[h == n, 0, (i + 1)*z*t^(i + j)/g[h + 1, n]]]; T[n_] := Function[p, Table[ Coefficient[p, t, k], {k, 0, Exponent[p, t]}]][SeriesCoefficient[ 1/g[0, n], {z, 0, n}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 07 2016, after Alois P. Heinz *)
f[i_] := If[i == 1, 1, -(i-1)^2 t^(2i - 3) z^2];
g[i_] := 1 - (2i - 1) t^(i-1) z;
cf = ContinuedFractionK[f[i], g[i], {i, 1, 5}];
CoefficientList[#, t]& /@ CoefficientList[cf + O[z]^10, z] // Rest // Flatten (* Jean-François Alcover, Nov 25 2018, after Mathieu Guay-Paquet *)

Prog

(Sage)
# The following Sage program
# yields the entries of the first n rows
# as a list of lists
def get_first_rows(n):
    R, t = PolynomialRing(ZZ, 't').objgen()
    S, z = PowerSeriesRing(R, 'z').objgen()
    gf = S(1).add_bigoh(1)
    for i in srange(n, 0, -1):
        a = (i+1) // 2
        b = i // 2
        gf = 1 / (1 - a * t^b * z * gf)
    return [list(row) for row in gf.shift(-1)]
# Mathieu Guay-Paquet, Apr 30 2014

Crossrefs

Cf. A005990, A009006, A062866, A062867, A062870, A072949, A301897.

Row sums give A000142.

A357329 is a sub-triangle.

Sequence in context: A152821 A071943 A357329 * A102473 A011117 A069269

Adjacent sequences: A062866 A062867 A062868 * A062870 A062871 A062872

Keyword

nonn,tabf,easy,look

Author

Olivier Gérard, Jun 26 2001

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Oct 03 2022

Status

approved