A048163 -id:A048163 - cp's OEIS frontend

A382676 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n+1,k+1)^2.

1, 4, 52, 1372, 60316, 3964684, 363503932, 44280657292, 6913081723516, 1345238707327564, 319137578070718012, 90648956570718822412, 30369040605677566161916, 11848724306426305222109644, 5325560174867275152102351292, 2731649923185995549312271694732

Offset: 0

Seiichi Manyama, Apr 03 2025

nonn

Main diagonal of A382673.

Cf. A048163, A092552, A382678.

a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n+1, k+1, 2)^2);

a(n) = (n!)^2 * [(x*y)^n] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

A213977 Number of n X n matrices with entries 0 and 1 and no 2 X 2 submatrix of form [ 1 1; 0 0 ].

Original entry on oeis.org

1, 2, 14, 200, 3536, 67472, 1423168, 34048352, 927156224, 28490354432, 976839578624, 36983803914752, 1532587515049984, 68997562105014272, 3353462146559209472, 175003916852177604608, 9760034505494167420928, 579311442062239341412352, 36462558160899681920745472, 2425761875540844266778656768

Offset: 0

N. J. A. Sloane, Jul 05 2012

nonn

Joerg Arndt, Table of n, a(n) for n = 0..100
Hyeong-Kwan Ju, Seunghyun Seo, Enumeration of 0/1-matrices avoiding some 2x2 matrices, arXiv:1107.1299 [math.CO], 2011.
H.-K. Ju and S. Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.

Cf. A048163, A014235.

terms = 20; w = ProductLog[-x E^x]; CoefficientList[-2w/(x(w+1)) + (x^2-1) E^(2x) - 2x(x+1) E^(4x) + O[x]^terms, x]*Range[0, terms-1]! (* Jean-François Alcover, Jul 22 2018, after Joerg Arndt *)

N=66; x='x+O('x^N);
W(x)=sum(n=1,N, (-n)^(n-1)*x^n/n! );
w=W(-x*exp(x));
egf=-2*w/(x*(1+w)) + (x^2-1)*exp(2*x)-2*x*(x+1)*exp(4*x);
Vec(serlaplace(egf))
/* Joerg Arndt, Jul 19 2012 */

Ju and Seo give an e.g.f. (see PARI code).

More terms from Joerg Arndt, Jul 19 2012

A297077 Number of distinct row and column sums for n X n (0, 1)-matrices.

Original entry on oeis.org

1, 2, 15, 328, 16145, 1475856, 219682183, 48658878080, 15047189968833, 6199170628499200, 3283463201858585471, 2174417430787021427712, 1760550428764505109190225, 1711145965399957937819322368, 1966168979042910307305159432375, 2636533865690624357354875499216896

Offset: 0

Peter Kagey, Dec 25 2017

nonn

a(n) is bounded above by 2^(n^2) and bounded below by A048163(n + 1).
Also the number of acyclic edge sets of the complete bipartite graph K_{n,n}. See proof by David E. Speyer at the Mathematics Stack Exchange link below.
It is also the number of n X n binary matrices that can be completed to the all-ones matrix by repeatedly changing an element from a zero to a one if that element is the only zero in its row or column. (Proof idea: Every acyclic edge set can be reduced to the empty set by removing one leaf edge at a time.) This can be interpreted as the number of ways of turning off storage nodes in an n X n array so that data remains restorable in the "full scheme" RAID (Redundant Array of Inexpensive Disks) design described by Nagel, Süß.- Jair Taylor, Jul 29 2019

			For n = 3, both of the following 3 X 3 (0,1)-matrices have identical row and column sums:
[1 0 1]     [1 1 0]
[0 1 0] and [0 1 0]
[0 1 0]     [0 0 1]
have row sums of [2 1 1] and column sums of [1 2 1].
So ([2 1 1], [1 2 1]) is one of the 328 possible row/column sums for a 3 X 3 matrix.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Mathematics Stack Exchange, Spanning forests of bipartite graphs and distinct row/column sums of binary matrices
Lars Nagel & Tim Süß, Computing the Probability for Data Loss in Two-Dimensional Parity RAIDs, 2017 13th European Dependable Computing Conference (EDCC), 58-65.
Rebecca J. Stones, Computing the number of h-edge spanning forests in complete bipartite graphs, Discrete Mathematics & Theoretical Computer Science, vol. 16, no. 1, pp. 313-326, 2014.

Main diagonal of A328887.

Cf. A048163 gives the number of row/column sums that uniquely identify a matrix.

{1}~Join~Array[Length@ Union@ Map[Total /@ {Transpose@ #, #} &, Tuples[{0, 1}, {#, #}]] &, 4] (* Michael De Vlieger, Jan 11 2018 *)

Terms a(6) and beyond from Andrew Howroyd, Oct 29 2019

A382678 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n+1,k+1)^2.

Original entry on oeis.org

1, 5, 77, 2357, 118061, 8712245, 886143917, 118592620277, 20176999414061, 4249819031692085, 1084956766012858157, 329975948760472311797, 117851658189070970988461, 48830366210401091606537525, 23228207308210113849419226797, 12571433948267218576823401692917

Offset: 0

Seiichi Manyama, Apr 03 2025

nonn

Main diagonal of A382674.

Cf. A048163, A092552, A382676.

a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n+1, k+1, 2)^2);

a(n) = (n!)^2 * [(x*y)^n] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.

A306267 Number of permutations of [n] within distance floor(n/2) of a fixed permutation.

Original entry on oeis.org

1, 1, 2, 3, 14, 31, 230, 675, 6902, 25231, 329462, 1441923, 22934774, 116914351, 2193664790, 12764590275, 276054834902, 1805409270031, 44222780245622, 321113303226243, 8787513806478134, 70146437009397871, 2121181056663291350, 18462286083671614275

Offset: 0

Alois P. Heinz, Feb 01 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..475

Cf. A306209.

Shifted bisections give: A048163 (even part), A092552 (odd part).

a(n) = A306209(n,floor(n/2)).

A382826 a(n) = Sum_{k=0..n} (k! * Stirling1(n+1,k+1))^2.

Original entry on oeis.org

1, 2, 17, 337, 12152, 696076, 58136500, 6673107316, 1008077743552, 193915431216576, 46281189562936704, 13420575661095930240, 4647502230640182602496, 1894412230202331489632256, 897850527136410029486517504, 489578762044356075253626875136

Offset: 0

Seiichi Manyama, Apr 06 2025

nonn

Main diagonal of A382823.

Cf. A048163, A382792.

a(n) = sum(k=0, n, (k!*stirling(n+1, k+1, 1))^2);

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y)) ).

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1+x) * (1+y) * (1 - log(1+x) * log(1+y)) ).

A143381 Number of Hi-Lo arrangements HL(m,n) of a deck with n suits and m ranks in each suit, m>=1, n>=1.

Original entry on oeis.org

0, 2, 0, 6, 2, 0, 14, 30, 2, 0, 78, 230, 174, 2, 0, 230, 14094, 4834, 1092, 2, 0, 1902, 187106, 3785126, 114442, 7188, 2, 0, 6902, 26185806, 250560122, 1225289412, 2908990, 48852, 2, 0, 76110, 557115782, 682502468094, 423419180642

Offset: 1

Max Alekseyev, Aug 11 2008, Aug 17 2008

In High-Low card game, a card is turned over (from the top of a regular shuffled 52-card deck) and the player is asked to guess if the next card will be higher or lower than the one shown. A simple strategy to play the game would be to guess 'High' if the card is an Ace through 6 (consider Ace to be of rank 1), 'Low' if the card is 8 through 13 (King) and flip a coin if the card is a 7. Intuitively, the player is playing the best he can without memory. If we make the assumption that the player always gets the random coin flips correct, then the probability that he will get every turn correct through the entire deck equals HL(13,4)*4!^13/52! (~= 1.7*10^(-7)) where HL(m,n) is defined below.

Given a deck with n suits each ranked from 1 to m (for a total of mn cards in the deck), a Hi-Lo arrangement of the cards is an arrangement of ranks r(1),r(2),...,r(mn) that satisfies the following three properties: (i) if r(i) < (m+1)/2 then r(i+1) > r(i); (ii) if r(i) > (m+1)/2 then r(i+1) < r(i); and (iii) if r(i) = (m+1)/2 then r(i+1) is different from r(i). The number of Hi-Lo arrangements of a deck with m ranks and n suits is denoted HL(m,n).

			The table of values HL(m,n) starts:
0 0 0 0 0 0 0 ...
2 2 2 2 2 2 2 ...
6 30 174 1092 7188 48852 339720 ...
14 230 4834 114442 2908990 77538470 2138286650 ...
78 14094 3785126 1225289412 442227602892 171398421245988 69859403814893544 ...
...

Max Alekseyev, PARI scripts for various problems
Kipli's Cage: Enumerating HiLo arrangements (the definition there has some glitches - see correct version in this entry).

Rows: A000004, A007395, A110706. Bisection of the first column: HL(2m, 1) = A048163(m+1).

\r nseqadj.gp
{ f(m,n,k) = sum(j=0, k, (-1)^j * binomial(k,j) * binomial(k-j,n)^m ) }
{ HL0(m,n) = 2 * sum(k=n, (m/2)*n, f(m/2,n,k) * (f(m/2,n,k) + f(m/2,n,k+1)) ) } \\ for even m
{ HL1(m,n) = sum(i=n, (m\2)*n, f(m\2,n,i) * sum(j=n, (m\2)*n, f(m\2,n,j) * M([n,i,j]) )) } \\ for odd m
{ HL(m,n) = if(m%2, HL1(m,n), HL0(m,n) ) }

cp's OEIS Frontend

A382676 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n+1,k+1)^2.

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A213977 Number of n X n matrices with entries 0 and 1 and no 2 X 2 submatrix of form [ 1 1; 0 0 ].

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Mathematica

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A297077 Number of distinct row and column sums for n X n (0, 1)-matrices.

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Mathematica

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A382678 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n+1,k+1)^2.

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PARI

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A306267 Number of permutations of [n] within distance floor(n/2) of a fixed permutation.

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A382826 a(n) = Sum_{k=0..n} (k! * Stirling1(n+1,k+1))^2.

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A143381 Number of Hi-Lo arrangements HL(m,n) of a deck with n suits and m ranks in each suit, m>=1, n>=1.

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PARI