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A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n). 14
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 46, 46, 16, 1, 1, 32, 146, 230, 146, 32, 1, 1, 64, 454, 1066, 1066, 454, 64, 1, 1, 128, 1394, 4718, 6902, 4718, 1394, 128, 1, 1, 256, 4246, 20266, 41506, 41506, 20266, 4246, 256, 1, 1, 512, 12866, 85310, 237686, 329462, 237686, 85310, 12866, 512, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]

B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]

T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Beáta Bényi, Advances in Bijective Combinatorics, Ph. D. Dissertation, Doctoral School of Mathematics and Computer Science, University of Szeged, Bolyai Institute, 2014. See Table 1.

Beáta Bényi, Peter Hajnal, Combinatorics of poly-Bernoulli numbers, arXiv:1510.05765 [math.CO], 2015.

Beata Bényi, Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016.

Beata Benyi, Peter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.

Beáta Bényi, Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], 2017.

Beáta Bényi, José Luis Ramírez, On q-poly-Bernoulli numbers arising from combinatorial interpretations, arXiv:1909.09949 [math.CO], 2019.

Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.

Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.

Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.

Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.

Stéphane Launois, Combinatorics of H-primes in quantum matrices, Journal of Algebra, Volume 309, Issue 1, 2007, Pages 139-167.

H. A. Witek, G. Mos, C.-P. Choua, Zhang-Zhang, Polynomials of Regular 3-and 4-tier Benzenoid Strips, MATCH Commun. Math. Comput. Chem. 73 (2015) 427-442.

FORMULA

pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ].

E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].

T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna, Nov 04 2004

n-th row of the array = row sums of n-th power of triangle A210381. - Gary W. Adamson, Mar 21 2012

EXAMPLE

1,  1,   1,    1,     1,      1, ...

1,  2,   4,    8,    16,     32, ...

1,  4,  14,   46,   146,    454, ...

1,  8,  46,  230,  1066,   4718, ...

1, 16, 146, 1066,  6902,  41506, ...

1, 32, 454, 4718, 41506, 329462, ...

...

MAPLE

A:= (n, k)-> add(Stirling2(n+1, i+1)*Stirling2(k+1, i+1)*

             i!^2, i=0..min(n, k)):

seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jan 02 2016

MATHEMATICA

T[n_, k_] := Sum[(-1)^(j+n)*(1+j)^k*j!*StirlingS2[n, j], {j, 0, n}]; Table[ T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)

PROG

(PARI) T(n, k)=sum(j=0, n, (j+1)^k*sum(i=0, j, (-1)^(n+j-i)*binomial(j, i)*(j-i)^n))

(PARI) T(n, k)=sum(j=0, min(n, k), j!^2*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2)); \\ Michel Marcus, Mar 05 2017

CROSSREFS

Rows 0-9 are A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.

Main diagonal is A048163. Another diagonal is A188634.

Antidiagonal sums are in A098830.

Cf. A019538, A210381, A266695.

Sequence in context: A154867 A064298 A256894 * A255256 A299906 A117401

Adjacent sequences:  A099591 A099592 A099593 * A099595 A099596 A099597

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Oct 27 2004

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)