A276975 -id:A276975 - cp's OEIS frontend

A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879, 32295634916658495460, 710503968166486900119

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the number of permutations in the symmetric group S_n that have a fixed point, i.e., they are not derangements (A000166). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001
a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=k! for k=0,1,...,n. - Michael Somos, Oct 07 2003
The termwise sum of this sequence and A000166 gives the factorial numbers. - D. G. Rogers, Aug 26 2006, Jan 06 2008
a(n) is the number of deco polyominoes of height n and having in the last column an odd number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the horizontal domino is the only deco polyomino of height 2 having an odd number of cells in the last column. - Emeric Deutsch, May 08 2008
Starting (1, 4, 15, 76, 455, ...) = eigensequence of triangle A127899 (unsigned). - Gary W. Adamson, Dec 29 2008
(n-1) | a(n), hence a(n) is never prime. - Jonathan Vos Post, Mar 25 2009
a(n) is the number of permutations of [n] that have at least one fixed point = number of positive terms in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
Numerator of partial sum of alternating harmonic series, provided that the denominator is n!. - Richard Locke Peterson, May 11 2020
a(n) is the number of terms in the polynomial expansion of the determinant of a n X n matrix that contains at least one diagonal element. - Adam Wang, May 28 2025

			G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ...

R. K. Guy, Unsolved Problems Number Theory, E37.
R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, Feb 10 1993
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
R. K. Guy and S. Washburn, Correspondence, Nov. - Dec. 1991
T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Exact formulas for Integer Sequences.
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraphs 5 and 7.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002468, A002469, A028306, A047920, A052169, A127899, A276975.
Row sums of A068106.
Column k=1 of A293211.
Column k=0 of A299789, A306234, and of A324362.

a := proc(n) -add((-1)^i*binomial(n, i)*(n-i)!, i=1..n) end;
a := n->-n!*add((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007
a := n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)):
seq(a(n), n=0..20); # Peter Luschny, Feb 28 2017

Denominator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* Wouter Meeussen, Mar 24 2007 *)
a[ n_] := If[ n < 0, 0, n! - Subfactorial[n]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 1, 0, n! - Round[ n! / E]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! - (-1)^n HypergeometricPFQ[ {- n, 1}, {}, 1]](* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - Exp[ -x] ) / (1 - x), {x, 0, n}]] (* Michael Somos, Jan 25 2014 *)
RecurrenceTable[{a[n] == (n - 1) ( a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)

{a(n) = if( n<1, 0, n * a(n-1) - (-1)^n)} /* Michael Somos, Mar 24 2003 */

{a(n) = if( n<0, 0, n! * polcoeff( (1 - exp( -x + x * O(x^n))) / (1 - x), n))} /* Michael Somos, Mar 24 2003 */

a(n) = if(n<1,0,subst(polinterpolate(vector(n,k,(k-1)!)),x,n+1))

A002467(n) = if(n<1, 0, n*A002467(n-1)-(-1)^n); \\ Joerg Arndt, Apr 22 2013

a(n) = n! - A000166(n) = A000142(n) - A000166(n).
E.g.f.: (1 - exp(-x)) / (1 - x). - Michael Somos, Aug 11 1999
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1; a(1) = 1. - Michael Somos, Aug 11 1999
a(n) = n*a(n-1) - (-1)^n. - Michael Somos, Aug 11 1999
a(0) = 0, a(n) = floor(n!(e-1)/e + 1/2) for n > 0. - Michael Somos, Aug 11 1999
a(n) = - n! * Sum_{i=1..n} (-1)^i/i!. Limit_{n->infinity} a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004
Inverse binomial transform of A002627. - Ross La Haye, Sep 21 2004
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1. - Gary Detlefs, Apr 11 2010
a(n) = n! - floor((n!+1)/e), n > 0. - Gary Detlefs, Apr 11 2010
For n > 0, a(n) = {(1-1/exp(1))*n!}, where {x} is the nearest integer. - Simon Plouffe, conjectured March 1993, added Feb 17 2011
0 = a(n) * (a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2) * (a(n+2)) if n >= 0. - Michael Somos, Jan 25 2014
a(n) = Gamma(n+1) - Gamma(n+1, -1)*exp(-1). - Peter Luschny, Feb 28 2017
a(n) = Sum_{k=0..n-1} A047920(n-1,k). - Alois P. Heinz, Sep 01 2021

A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

Original entry on oeis.org

0, 1, 3, 13, 67, 411, 2921, 23633, 214551, 2160343, 23897269, 288102189, 3760013027, 52816397219, 794536751217, 12744659120521, 217140271564591, 3916221952414383, 74539067188152941, 1493136645424092773, 31400620285465593339, 691708660911435955579

Offset: 1

Emeric Deutsch, Sep 07 2010

nonn

a(n) = A180190(n,1).
a(n+2) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = k! for k = 1, ..., n+1. - Michael Somos, Jan 05 2012
From Jon Perry, Jan 04 2013: (Start)
Number of permutations of {1,...,n-1,n+1} with at least one indexed point p(k)=k with 1<=k<=n. Note that this means p(k)=n+1 is never an indexed point as kFor n>1, a(n) is the number of permutations of [n+1] that have a fixed point and contain 12; for example the a(3)=3 such permutations of {1,2,3,4} are 1234, 1243, and 3124.
(End)
For n > 0: row sums of triangle A116853. - Reinhard Zumkeller, Aug 31 2014

			x^2 + 3*x^3 + 13*x^4 + 67*x^5 + 411*x^6 + 2921*x^7 + 23633*x^8 + ...
a(3) = 3 because we have 123, 312, and 231; the permutations 132, 213, and 321 have no successions.
a(4) = 13 since p(x) = (3*x^2 - 7*x + 6) / 2 interpolates p(1) = 1, p(2) = 2, p(3) = 6, and p(4) = 13. - _Michael Somos_, Jan 05 2012

Alois P. Heinz, Table of n, a(n) for n = 1..200
Uriel Feige, Tighter bounds for online bipartite matching, 2018.

Cf. A000142, A000166, A000255, A002467, A028417, A180190.
Cf. A116853, A276975.
Column k=1 of A306234, A306461, and of A324362(n-1).

a180191 n = if n == 1 then 0 else sum $ a116853_row (n - 1)
-- Reinhard Zumkeller, Aug 31 2014

d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: seq(factorial(n)-d[n]-d[n-1], n = 1 .. 22);

f[n_] := Sum[ -(-1)^k (n - k)! Binomial[n - 1, k], {k, 1, n}]; Array[f, 20] (* Robert G. Wilson v, Oct 16 2010 *)

{a(n) = if( n<2, 0, n--; subst( polinterpolate( vector( n, k, k!)), x, n+1))} /* Michael Somos, Jan 05 2012 */

a(n) = n! - d(n) - d(n-1), where d(j) = A000166(j) are the derangement numbers.
a(n) = n! - A000255(n-1) = A002467(n) - A000166(n-1). - Jon Perry, Jan 05 2013
a(n) = (n-1)! [x^(n-1)] (1-exp(-x))/(1-x)^2. - Alois P. Heinz, Feb 23 2019

A276974 Number T(n,k) of permutations of [n] where the minimal distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 1, 1, 0, 19, 3, 1, 1, 0, 103, 12, 3, 1, 1, 0, 651, 54, 10, 3, 1, 1, 0, 4702, 281, 42, 10, 3, 1, 1, 0, 38413, 1652, 203, 37, 10, 3, 1, 1, 0, 350559, 11017, 1086, 166, 37, 10, 3, 1, 1, 0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1, 0, 39196758, 669948, 44265, 4900, 726, 151, 37, 10, 3, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2016

			T(3,1) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).
T(3,2) = 1: (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,     1;
  0,       4,     1,    1;
  0,      19,     3,    1,   1;
  0,     103,    12,    3,   1,   1;
  0,     651,    54,   10,   3,   1,  1;
  0,    4702,   281,   42,  10,   3,  1,  1;
  0,   38413,  1652,  203,  37,  10,  3,  1, 1;
  0,  350559, 11017, 1086, 166,  37, 10,  3, 1, 1;
  0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Columns k=0-1 give: A000007, A276975.
Row sums give A000142.
T(2n,n) = A138378(n) = A005493(n-1) for n>0.
Cf. A239145, A263757, A277031.

A277032 Number of permutations of [n] such that the minimal cyclic distance between elements of the same cycle equals one, a(1)=1 by convention.

Original entry on oeis.org

1, 1, 5, 20, 109, 668, 4801, 38894, 353811, 3561512, 39374609, 474132730, 6179650125, 86676293916, 1301952953989, 20852719565694, 354771488612075, 6389625786835184, 121456993304945749, 2429966790591643402, 51042656559451380013, 1123165278137918510772

Offset: 1

Alois P. Heinz, Sep 25 2016

nonn

			a(2) = 1: (1,2).
a(3) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).

Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Column k=1 of A277031.

Cf. A002467, A180191, A276975.

b:= proc(n, i, l) option remember; `if`(n=0, mul(j!, j=l),
      (m-> add(`if`(i=j or n*j=1, 0, b(n-1, j, `if`(j>m,
      [l[], 0], subsop(j=l[j]+1, l)))), j=1..m+1))(nops(l)))
    end:
a:= n-> `if`(n=1, 1, n!-b(n-1, 1, [0])):
seq(a(n), n=1..15);

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Product[j!, {j, l}], With[{m = Length[l]}, Sum[If[i == j || n*j == 1, 0, b[n-1, j, If[j>m, Append[l, 0], ReplacePart[l, j -> l[[j]]+1]]]], {j, 1, m+1}]]];
a[n_] := If[n == 1, 1, n! - b[n-1, 1, {0}]];
Array[a, 15] (* Jean-François Alcover, Mar 13 2021, after Alois P. Heinz *)

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cp's OEIS Frontend

A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

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A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

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A276974 Number T(n,k) of permutations of [n] where the minimal distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A277032 Number of permutations of [n] such that the minimal cyclic distance between elements of the same cycle equals one, a(1)=1 by convention.

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