A129178 - OEIS

%I #24 Nov 03 2025 13:12:57

%S 1,1,2,4,2,8,8,6,2,16,24,28,26,16,8,2,32,64,96,120,126,110,82,52,26,

%T 10,2,64,160,288,432,564,658,680,638,542,416,284,172,90,38,12,2,128,

%U 384,800,1376,2072,2824,3526,4058,4344,4346,4066,3562,2912,2218,1566,1016,598

%N Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that invc(p)=k (n >= 0; 0 <= k <= (n-1)(n-2)/2) (see comment for invc definition).

%C invc(p) is defined (by Carlitz) in the following way: express p in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then remove the parentheses and count the inversions in the obtained word.

%C Row n has 1+(n-1)*(n-2)/2 - delta_{0,n} terms. Row sums are the factorials (A000142). T(n,0) = 2^(n-1) = A011782(n) = A000079(n-1). T(n,1) = (n-2)*2^(n-2) = A036289(n-2) for n>=2. T(n,k) = A121552(n,n+k).

%C It appears that Sum_{k>=0} k*T(n,k) = A126673(n).

%D L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University, 1968, 1-7.

%H Alois P. Heinz, <a href="/A129178/b129178.txt">Rows n = 0..50, flattened</a>

%H Toufik Mansour, Mark Shattuck, <a href="http://doi.org/10.1007/s13370-012-0106-6">A q-analog of the hyperharmonic numbers</a>, Afrika Matematika 25.1 (2014): 147-160.

%H M. Shattuck, <a href="http://www.integers-ejcnt.org/f7/f7.Abstract.html">Parity theorems for statistics on permutations and Catalan words</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.

%F Generating polynomial of row n is P[n](t) = 2*(2+t)*(2+t+t^2)*...*(2 + t + t^2 + ... + t^(n-2)) for n >= 3, P[1](t)=1, P[2](t)=2.

%e T(3,0)=4, T(3,1)=2 because we have 123=(1)(2)(3), 132=(1)(23), 213=(12)(3), 231=(123) with the resulting word (namely 123) having 0 inversions and 312=(132) and (321)=(13)(2) with the resulting word (namely 132) having 1 inversion.

%e Triangle starts:

%e    1;

%e    1;

%e    2;

%e    4,   2;

%e    8,   8,   6,   2;

%e   16,  24,  28,  26,  16,   8,   2;

%e   32,  64,  96, 120, 126, 110,  82,  52,  26,  10,  2;

%e   ...

%p s:=j->2+sum(t^i, i=1..j): for n from 0 to 9 do P[n]:=sort(expand(simplify(product(s(j), j=0..n-2)))) od: for n from 0 to 9 do seq(coeff(P[n], t, j), j=0..degree(P[n])) od;  # yields sequence in triangular form

%t nMax = 9; s[j_] := 2 + Sum[t^i, {i, 1, j}]; P[0] = P[1] = 1; P[2] = 2; For[ n = 3, n <= nMax, n++, P[n] = Sort[Expand[Simplify[Product[s[j], {j, 0, n-2}]]]]]; Table[Coefficient[P[n], t, j], {n, 0, nMax}, {j, 0, Exponent[ P[n], t]}] // Flatten (* _Jean-François Alcover_, Jan 24 2017, adapted from Maple *)

%Y Cf. A000142, A011782, A000079, A036289, A121552, A126673.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Apr 11 2007

%E One term for row n=0 prepended by _Alois P. Heinz_, Dec 16 2016