A216120 Irregular triangle read by rows: T(n,k) is the number of permutations in S_n having k stretching pairs.

1, 2, 6, 22, 2, 94, 22, 4, 462, 172, 72, 12, 2, 2582, 1244, 824, 276, 94, 16, 4, 16214, 9126, 8016, 3996, 1990, 660, 248, 56, 12, 2, 113166, 70482, 74220, 48012, 30898, 14372, 7520, 2720, 1068, 318, 84, 16, 4, 869662, 581264, 690744, 534000, 414532, 239704, 156440, 75668, 39256, 16952, 7032, 2384, 868, 224, 56, 12, 2

Offset: 1

Author

Emeric Deutsch, Feb 26 2013

Keywords

• nonn
• tabf

Comments

A stretching pair of a permutation p in S_n is a pair (i,j) (1<=i < j<=n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because p(2) = 1 < 2 < 4 < p(4) = 5.

Sum of entries in row n is n! = A000142(n).

Sum(k*T(n,k), k>=1) = A216119(n).

Examples

			T(4,1) = 2 because 2143 has 1 stretching pair (2,3) and 3142 has 1 stretching pair (2,3); the other 22 permutations in S_4 have no stretching pairs.
Triangle starts:
1;
2;
6;
22,      2;
94,     22,   4;
462,   172,  72,  12,  2;
2582, 1244, 824, 276, 94, 16, 4;
		

References

• E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted, 2013)

Links

• E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31, 2010, 270-279.
• J. Cooper, E. Lundberg, and B. Nagle, Generalized pattern frequency in large permutations, Electron. J. Combin. 20, 2013, #P28.

Crossrefs

Cf. A000142, A216119, A216121.

Programs

Maple

n := 7: with(combinat): sp := proc (p) local ct, i, j: ct := 0: for i from 2 to nops(p)-2 do for j from i+1 to nops(p)-1 do if p[i] < i and i < j and j < p[j] then ct := ct+1 else  end if end do end do: ct end proc: P := permute(n): f[n] := sort(add(t^sp(P[j]), j = 1 .. factorial(n)));

Formula

The values of T(n,k) have been found by straightforward counting (with Maple). The Maple program yields the generating polynomial of the specified row n. Within the program, sp(p) is the number of stretching pairs of the permutation p.