Search: a052571 -id:a052571
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A005990
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a(n) = (n-1)*(n+1)!/6.
(Formerly M4551)
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+10
21
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0, 1, 8, 60, 480, 4200, 40320, 423360, 4838400, 59875200, 798336000, 11416204800, 174356582400, 2833294464000, 48819843072000, 889218570240000, 17072996548608000, 344661117825024000, 7298706024529920000, 161787983543746560000
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OFFSET
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1,3
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COMMENTS
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Coefficients of Gandhi polynomials.
a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0), i.e., the total positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006
a(n) is also the sum of the excedances of all permutations of [n]. An excedance of a permutation p of [n] is an i (1 <= i <= n-1) such that p(i) > i. Proof: i is an excedance if p(i) = i+1, i+2, ..., n (n-i possibilities), with the remaining values of p forming any permutation of [n]\{p(i)} in the positions [n]\{i} ((n-1)! possibilities). Summation of i(n-i)(n-1)! over i from 1 to n-1 completes the proof. Example: a(3)=8 because the permutations 123, 132, 213, 231, 312, 321 have excedances NONE, {2}, {1}, {1,2}, {1}, {1}, respectively. - Emeric Deutsch, Oct 26 2008
a(n) is also the number of doubledescents in all permutations of {1,2,...,n-1}. We say that i is a doubledescent of a permutation p if p(i) > p(i+1) > p(i+2). Example: a(3)=8 because each of the permutations 1432, 4312, 4213, 2431, 3214, 3421 has one doubledescent, the permutation 4321 has two doubledescents and the remaining 17 permutations of {1,2,3,4} have no doubledescents. - Emeric Deutsch, Jul 26 2009
Half of sum of abs(p(i+1) - p(i)) over all permutations on n, e.g., 42531 = 2 + 3 + 2 + 2 = 9, and the total over all permutations on {1,2,3,4,5} is 960. - Jon Perry, May 24 2013
a(n) gives the number of non-occupied corners in tree-like tableaux of size n+1 (see Gao et al. link). - Michel Marcus, Nov 18 2015
a(n) is the number of sequences of n+2 balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 26 2017
a(n) is the number of triangles (3-cycles) in the (n+1)-alternating group graph. - Eric W. Weisstein, Jun 09 2019
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alice L. L. Gao, Emily X. L. Gao, and Brian Y. Sun, Zubieta's Conjecture on the Enumeration of Corners in Tree-like Tableaux, arXiv:1511.05434 [math.CO], 2015. The second version of this paper has a different title and different authors: A. L. L. Gao, E. X. L. Gao, P. Laborde-Zubieta, and B. Y. Sun, Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue, arXiv preprint arXiv:1511.05434v2, 2015.
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FORMULA
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a(n) = Sum_{m=0..n} Sum_{k=-1..n} Sum_{j=1..n} n!/6, n >= 0. - Zerinvary Lajos, May 11 2007
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j)) then a(n+1) = (-1)^(n-1)*f(n,1,-4), (n >= 1). - Milan Janjic, Mar 01 2009
E.g.f.: (-1+3*x)/(3!*(1-x)^3), a(0) = -1/3!. Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010
a(n) = ((n+3)!/2) * Sum_{j=i..k} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010
a(n) = (n+2)!*Sum_{k=1..n-1} 1/((2*k+4)*(k+3)). - Gary Detlefs, Oct 09 2011
a(n) = (n+2)!*(1 + 3*(H(n+1) - H(n+2)))/6, where H(n) is the n-th harmonic number. - Gary Detlefs, Oct 09 2011
Sum_{n>=2} 1/a(n) = 3*(Ei(1) - gamma) - 6*e + 27/2, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=2} (-1)^n/a(n) = 3*(gamma - Ei(-1)) - 3/2, where Ei(-1) = -A099285. (End)
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MAPLE
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[ seq((n-1)*(n+1)!/6, n=1..40) ];
a:=n->sum(sum(sum(n!/6, j=1..n), k=-1..n), m=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, May 11 2007
seq(sum(mul(j, j=3..n), k=3..n)/3, n=2..21); # Zerinvary Lajos, Jun 01 2007
restart: G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=2..21); # Zerinvary Lajos, Apr 01 2009
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MATHEMATICA
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Range[0, 20]! CoefficientList[Series[x/(1 - x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Better definition from Robert Newstedt
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STATUS
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approved
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A257503
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Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).
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+10
16
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1, 2, 4, 3, 12, 18, 5, 16, 72, 96, 6, 22, 90, 480, 600, 7, 48, 114, 576, 3600, 4320, 8, 52, 360, 696, 4200, 30240, 35280, 9, 60, 378, 2880, 4920, 34560, 282240, 322560, 10, 64, 432, 2976, 25200, 39600, 317520, 2903040, 3265920, 11, 66, 450, 3360, 25800, 241920, 357840, 3225600, 32659200, 36288000, 13, 70, 456, 3456, 28800, 246240, 2540160, 3588480, 35925120, 399168000, 439084800
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OFFSET
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1,2
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COMMENTS
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The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The first row (A256450) contains all the numbers which have at least one 1-digit in their factorial base representation (see A007623), after which the successive rows are obtained from the terms on the row immediately above by shifting their factorial representation one left and then incrementing the nonzero digits in that representation with a factorial base shift-operation A255411.
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LINKS
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FORMULA
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A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)).
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EXAMPLE
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The top left corner of the array:
1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13
4, 12, 16, 22, 48, 52, 60, 64, 66, 70, 76
18, 72, 90, 114, 360, 378, 432, 450, 456, 474, 498
96, 480, 576, 696, 2880, 2976, 3360, 3456, 3480, 3576, 3696
600, 3600, 4200, 4920, 25200, 25800, 28800, 29400, 29520, 30120, 30840
4320, 30240, 34560, 39600, 241920, 246240, 272160, 276480, 277200, 281520, 286560
...
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PROG
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(Scheme)
(define (A257503bi row col) (if (= 1 row) (A256450 (- col 1)) (A255411 (A257503bi (- row 1) col))))
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CROSSREFS
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Column 4: A213167 (without the initial one).
Column 5: A052571 (without initial zeros).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A257505
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Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.
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+10
15
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1, 4, 2, 18, 12, 3, 96, 72, 16, 5, 600, 480, 90, 22, 6, 4320, 3600, 576, 114, 48, 7, 35280, 30240, 4200, 696, 360, 52, 8, 322560, 282240, 34560, 4920, 2880, 378, 60, 9, 3265920, 2903040, 317520, 39600, 25200, 2976, 432, 64, 10, 36288000, 32659200, 3225600, 357840, 241920, 25800, 3360, 450, 66, 11, 439084800, 399168000, 35925120, 3588480, 2540160, 246240, 28800, 3456, 456, 70, 13
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OFFSET
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1,2
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COMMENTS
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The array is read by downward antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
In Kimberling's terminology, this array is called the dispersion of sequence A255411 (when started from its first nonzero term, 4). The left column is the complement of that sequence, which is A256450.
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LINKS
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FORMULA
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A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)).
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EXAMPLE
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The top left corner of the array:
1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920
2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200
3, 16, 90, 576, 4200, 34560, 317520, 3225600, 35925120
5, 22, 114, 696, 4920, 39600, 357840, 3588480, 39553920
6, 48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200
7, 52, 378, 2976, 25800, 246240, 2575440, 29352960, 362517120
8, 60, 432, 3360, 28800, 272160, 2822400, 31933440, 391910400
9, 64, 450, 3456, 29400, 276480, 2857680, 32256000, 395176320
10, 66, 456, 3480, 29520, 277200, 2862720, 32296320, 395539200
11, 70, 474, 3576, 30120, 281520, 2898000, 32618880, 398805120
13, 76, 498, 3696, 30840, 286560, 2938320, 32981760, 402433920
14, 84, 552, 4080, 33840, 312480, 3185280, 35562240, 431827200
15, 88, 570, 4176, 34440, 316800, 3220560, 35884800, 435093120
17, 94, 594, 4296, 35160, 321840, 3260880, 36247680, 438721920
19, 100, 618, 4416, 35880, 326880, 3301200, 36610560, 442350720
20, 108, 672, 4800, 38880, 352800, 3548160, 39191040, 471744000
21, 112, 690, 4896, 39480, 357120, 3583440, 39513600, 475009920
23, 118, 714, 5016, 40200, 362160, 3623760, 39876480, 478638720
...
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PROG
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(Scheme)
(define (A257505bi row col) (if (= 1 col) (A256450 (- row 1)) (A255411 (A257505bi row (- col 1)))))
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CROSSREFS
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Row 4: A213167 (without the initial one).
Row 5: A052571 (without initial zeros).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A282466
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a(n) = n*a(n-1) + n!, with n>0, a(0)=5.
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+10
9
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5, 6, 14, 48, 216, 1200, 7920, 60480, 524160, 5080320, 54432000, 638668800, 8143027200, 112086374400, 1656387532800, 26153487360000, 439378587648000, 7825123418112000, 147254595231744000, 2919482409811968000, 60822550204416000000, 1328364496464445440000
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OFFSET
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0,1
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REFERENCES
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C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), page 240 (Example 9.57 gives the recurrence).
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LINKS
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FORMULA
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E.g.f.: (5 - 4*x)/(1 - x)^2.
a(n) = (n + 5)*n!.
Sum_{n>=0} 1/a(n) = 9*e - 24.
Sum_{n>=0} (-1)^n/a(n) = 24 - 65/e. (End)
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MATHEMATICA
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RecurrenceTable[{a[0] == 5, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}] (* or *)
Table[(n + 5) n!, {n, 0, 30}]
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CROSSREFS
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Cf. sequences with formula (n + k)*n!: A052521 (k=-5), A282822 (k=-4), A052520 (k=-3), A052571 (k=-2), A062119 (k=-1), A001563 (k=0), A000142 (k=1), A001048 (k=2), A052572 (k=3), A052644 (k=4), this sequence (k=5).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000
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OFFSET
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1,2
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COMMENTS
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a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} |pi(i)-i|, i.e., the total displacement of all letters in all permutations on n letters.
a(n) = number of entries between the entries 1 and 2 in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 123, 1(3)2, 213, 2(3)1, 312, 321; the entries between 1 and 2 are surrounded by parentheses. - Emeric Deutsch, Apr 06 2008
a(n) is also the number of peaks in all permutations of {1,2,...,n+1}. Example: a(3)=16 because the permutations 1234, 4123, 3124, 4312, 2134, 4213, 3214, and 4321 have no peaks and each of the remaining 16 permutations of {1,2,3,4} has exactly one peak. - Emeric Deutsch, Jul 26 2009
a(n), for n>=2, is the number of (n+2)-node tournaments that have exactly one triad. Proven by Kadane (1966), see link. - Ian R Harris, Sep 26 2022
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REFERENCES
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D. Daly and P. Vojtechovsky, Displacement of permutations, preprint, 2003.
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LINKS
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FORMULA
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a(n) = (n+3)! * Sum_{k=1..n} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010
Sum_{n>=2} 1/a(n) = (3/2)*(Ei(1) - gamma) - 3*e + 27/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=2} (-1)^n/a(n) = (3/2)*(gamma - Ei(-1)) - 3/4, where Ei(-1) = -A099285. (End)
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MATHEMATICA
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nn=20; Drop[Range[0, nn]!CoefficientList[Series[ x^3/3/(1-x)^2, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Mar 04 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324225
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Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
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+10
3
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1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040
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OFFSET
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1,3
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COMMENTS
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T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].
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LINKS
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FORMULA
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T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise.
E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1.
|T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n.
Sum_{k=0..n-1} T(n,k) = A001710(n+1).
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EXAMPLE
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The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
[1 ] [1 ] [ 1 ] [ 1 ] [ 1] [ 1]
[ 1 ] [ 1] [1 ] [ 1] [1 ] [ 1 ]
[ 1] [ 1 ] [ 1] [1 ] [ 1 ] [1 ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
: 1 ;
: 1, 2, 1 ;
: 2, 4, 6, 4, 2 ;
: 6, 12, 18, 24, 18, 12, 6 ;
: 24, 48, 72, 96, 120, 96, 72, 48, 24 ;
: 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120 ;
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MAPLE
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b:= proc(s, c) option remember; (n-> `if`(n=0, c,
add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!, 0))(abs(k)):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
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MATHEMATICA
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T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!, 0]];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 25 2021, after 3rd Maple program *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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