1,7

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 Tanimoto reference.

Shinji Tanimoto,Alternate Permutations and Signed Eulerian Numbers, math.CO/0612135.

T(n,k) = T(n,n-k-1).

Triangle begins:

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].

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 (mathar(AT)strw.leidenuniv.nl), Dec 12 2006

Cf. A008292 = Triangle of Eulerian numbers T(n, k) read by rows, A049061 = Triangle a(n, k) (1<=k<=n) of signed Eulerian numbers.

Adjacent sequences: A125297 A125298 A125299 this_sequence A125301 A125302 A125303

Sequence in context: A107333 A098505 A021306 this_sequence A126717 A124039 A096433

nonn,tabl

Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 08 2006, Dec 11 2006

Corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 12 2006

Edited by njas, Dec 21 2006