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A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n). 16
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)

T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020

When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021

LINKS

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

Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, arXiv:2104.13654 [math.CO], 2021. See Table 1 (a) p. 4.

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

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

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

Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.

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

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

Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers - the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.

Beáta Bényi and José Luis Ramírez, Poly-Cauchy Numbers of the Second Kind-the Combinatorics Behind, Enumerative Comb. Appl. (2022) Vol. 2, No. 1, Art. #S2R1.

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

David Callan, Permutations whose excedance positions are those before 1

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.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

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

Wikipedia, Acyclic orientation

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 A328887 A299906

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 August 9 05:53 EDT 2022. Contains 356016 sequences. (Running on oeis4.)