A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1

Offset: 2

Author

N. J. A. Sloane, May 07 2016

Keywords

• nonn
• tabl

Comments

Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - N. J. A. Sloane, Jan 10 2018

Examples

			Triangle begins:
n\m  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]  1;
[3]  1,      1;
[4]  1,      5,      1;
[5]  1,     13,     13,      1;
[6]  1,     29,     73,     29,      1;
[7]  1,     61,    301,    301,     61,      1;
[8]  1,    125,   1081,   2069,   1081,    125,      1;
[9]  1,    253,   3613,  11581,  11581,   3613,    253,      1;
...
		

Links

• Gheorghe Coserea, Rows n = 2..101, flattened
• F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
• Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
• Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
• D. E. Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
• D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
• J. Riordan, Letter to N. J. A. Sloane, Dec. 1976.

Crossrefs

Column 2 is A036563.

Largest term in each row gives A272645.

Second diagonal from the right is 2^i - 3.

Third diagonal from the right edge is A006230.

T(2n,n) gives A048144.

For row sums see A297195.

Cf. A008277, A001469, A371761.

Programs

Maple

A272644 := proc(n,m)
    add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
end proc:
seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # R. J. Mathar, Mar 04 2018

Mathematica

Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten  (* Michael De Vlieger, May 19 2016 *)

PARI

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

Formula

T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.

A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016

Extensions

More terms from Gheorghe Coserea, May 16 2016