1,5

A "consecutive ascending pair" in a permutation p_1, p_2,..., p_n is a pair p_i, p_{i+1} = p_i + 1.

FromEmeric Deutsch,May 15 2010: (Start)

The same triangle, but with rows indexed differently, also arises as follows: U(n,k) = number of permutations of [n] having k blocks (1 <= k <= n), where a block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.

When seen as coefficients of polynomials with decreasing exponents: evaluations areA001339(x=2),A081923(x=3),A081924(x=4),A087981(x=-1).

The sum of the entries in row n is n!.

U(n,n) =A000255(n-1) = d(n-1) + d(n), U(n,n-1)=d(n), where d(j)=A000166(j) (derangement numbers). (End)

This is essentially the reversal of the exponential Riordan array [exp(-x)/(1-x)^2,x] (cf.A123513). -Paul Barry,Jun 17 2010

U(n-1, k-2) * n*(n-1)/k = number of permutations of [n] with k elements not fixed by the permutation. -Michael Somos,Aug 19 2018

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz,Rows n = 1..150, flattened

A. N. Myers,Counting permutations by their rigid patterns,J. Combin. Theory, A 99 (2002), 345-357. [Emeric Deutsch,May 15 2010]

E.g.f.: exp(x*(y-1))/(1-x)^2. -Vladeta Jovovic,Jan 03 2003

FromEmeric Deutsch,May 15 2010: (Start)

U(n,k) = binomial(n-1,k-1)*(k-1)!*Sum_{j=0..k-1} (-1)^(k-j-1)*(j+1)/(k-j-1)! (1 <= k <= n).

U(n,k) = (k+1)!*binomial(n,k)*(1/n)*Sum_{i=0..k+1} (-1)^i/i!.

U(n,k) = (1/n)*binomial(n,k)*d(k+1), where d(j)=A000166(j) (derangement numbers). (End)

Triangle starts:

1;

1, 1;

1, 2, 3;

1, 3, 9, 11;

1, 4, 18, 44, 53;

1, 5, 30, 110, 265, 309;

1, 6, 45, 220, 795, 1854, 2119;

1, 7, 63, 385, 1855, 6489, 14833, 16687;

1, 8, 84, 616, 3710, 17304, 59332, 133496, 148329;

1, 9, 108, 924, 6678, 38934, 177996, 600732, 1334961, 1468457;

...

For n=3, the permutations 123, 132, 213, 231, 312, 321 have respectively 2,0,0,1,1,0 consecutive ascending pairs, so row 3 of the triangle is 3,2,1. -N. J. A. Sloane,Apr 12 2014

In the alternative definition, T(4,2)=3 because we have 234.1, 4.123, and 34.12 (the blocks are separated by dots). -Emeric Deutsch,May 16 2010

U:= proc (n, k) options operator, arrow: factorial(k+1)*binomial(n, k)*(sum((-1)^i/factorial(i), i = 0.. k+1))/n end proc: for n to 10 do seq(U(n, k), k = 1.. n) end do; # yields sequence in triangular form; #Emeric Deutsch,May 15 2010

t[n_, k_]:= Binomial[n, k]*Subfactorial[k+1]/n; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (*Jean-François Alcover,Jan 07 2014, afterEmeric Deutsch*)

Diagonals, reading from the right-hand edge:A000255,A000166,A000274,A000313,A001260,A001261.A045943is another diagonal.

Cf.A123513(mirror image).

A289632is the analogous triangle with the additional restriction that all consecutive pairs must be isolated, i.e., must not be back-to-back to form longer consecutive sequences.

Sequence in context:A171150A111589A259760*A151880A108990A145080

Adjacent sequences:A010024A010025A010026*A010028A010029A010030

tabl,nonn

More terms fromVladeta Jovovic,Jan 03 2003

Original definition from David, Kendall and Barton restored byN. J. A. Sloane,Apr 12 2014

approved