OFFSET
1,8
COMMENTS
A permutation is a parity-alternating permutation (PAP) if its entries take even and odd integers alternately (examples: 436125, 563412 or 7216345). When n is an odd integer, odd entries must appear at both ends of PAPs. T(n,k) = the number of PAPs of {1,2,...,n} with exactly k ascents. Row sums = 2*((n/2)!)^2 if n is even and ((n+1)/2)!*((n-1)/2)! if n is odd.
Arises in combinatorial analysis of signed Eulerian numbers and parity-alternate permutations. This table is the first of three tables on p. 4 of the Tanimoto reference.
LINKS
Alois P. Heinz, Rows n = 1..27, flattened
Shinji Tanimoto, Alternate Permutations and Signed Eulerian Numbers, math.CO/0612135, Ann. Comb. 14 (2010), 355.
FORMULA
T(n,k) = T(n,n-k-1).
EXAMPLE
Triangle begins:
n=1.|.1
n=2.|.1....1
n=3.|.1....0....1
n=4.|.1....3....3....1
n=5.|.1....2....6....2....1
n=6.|.1....9...26...26....9....1
n=7.|.1....8...39...48...39....8....1
n=8.|.1...23..165..387..387..165...23....1
n=9.|.1...22..228..674.1030..674..228...22....1
n=10|.1...53..860.4292.9194.9194.4292..860...53....1
Examples of parity-alternating permutations of n=5 and their number of rises k are [1,2,3,4,5] (k=4, only rises), [1,2,5,4,3] (k=2: 1->2 and 2->5), [1,4,3,2,5] (k=2: 1->4 and 2->5). The T(n=5,k=1)=2 parity-alternating permutations with k=1 rise are [3,2,5,4,1] and [5,2,1,4,3].
MAPLE
isPAP := proc(per) local i ; for i from 2 to nops(per) do if ( op(i, per) mod 2 ) = (op(i-1, per) mod 2 ) then RETURN(false) ; fi ; od : RETURN(true) ; end: ascents := proc(per) local i, asc ; asc :=0 ; for i from 2 to nops(per) do if op(i, per) > op (i-1, per) then asc := asc+1 : fi ; od : RETURN(asc) ; end:
A125300row := proc(n) local per, resul, asc, thisp, p, i, row ; row := array(0..n-1) ; for i from 0 to n-1 do row[i] := 0 : od ; per := combinat[permute](n) ; for p from 1 to nops(per) do asc := 0 ; thisp := op(p, per) ; if isPAP(thisp) then asc := ascents(thisp) ; row[asc] := row[asc]+1 ; fi ; od ; RETURN(row) : end: for n from 2 to 10 do r := A125300row(n) ; for k from 0 to n-1 do print(r[k]) ; od : od : # R. J. Mathar, Dec 12 2006
MATHEMATICA
isPAP[per_] := (For[i = 2, i <= Length[per], i++, If [Mod[per[[i]], 2] == Mod[per[[i - 1]], 2], Return[False] ] ]; True);
ascents[per_] := (asc = 0; For[i = 2, i <= Length[per], i++, If[per[[i]] > per[[i - 1]], asc ++] ]; asc);
A125300row[n_] := (row = Range[0, n - 1]; For[i = 0, i <= n - 1, i++, row[[i]] = 0]; per = Permutations[Range[n]]; For[p = 1, p <= Length[per], p++, asc = 0; thisp = per[[p]]; If[isPAP[thisp], asc = ascents[thisp]; row[[asc]] += 1]]; row);
Join[{1}, Reap[For[n = 2, n <= 10, n++, r = A125300row[n]; For[k = 0, k <= n - 1, k++, Print[r[[k]]]; Sow[r[[k]]]]]][[2, 1]]] (* Jean-François Alcover, Nov 07 2017, after R. J. Mathar's Maple code *)
CROSSREFS
Cf. A008292 = Triangle of Eulerian numbers T(n, k) read by rows, A049061 = Triangle a(n, k) (1<=k<=n) of signed Eulerian numbers.
Row sums give: A092186. - Alois P. Heinz, Nov 18 2013
KEYWORD
nonn,tabl
AUTHOR
Jonathan Vos Post, Dec 08 2006, Dec 11 2006
EXTENSIONS
Corrected by R. J. Mathar, Dec 12 2006
Edited by N. J. A. Sloane, Dec 21 2006
Replaced arXiv URL by non-cached version - R. J. Mathar, Oct 30 2009
More terms from Alois P. Heinz, Nov 18 2013
STATUS
approved