A007049 -id:A007049

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A007075

Number of irreducible positions of size n in Montreal solitaire.
(Formerly M1441)

+10
6

1, 2, 5, 13, 35, 95, 260, 714, 1965, 5415, 14934, 41206, 113730, 313958, 866801, 2393315, 6608473, 18248017, 50389350, 139144906, 384237186, 1061044865, 2930013158, 8091077148, 22343115337, 61699480866, 170380367189, 470497972866

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) is also the number of indecomposable permutations with exactly n inversions; there is one indecomposable permutation with no inversions. - David Bevan, Dec 19 2017

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

David Bevan, Table of n, a(n) for n = 1..1000

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

FORMULA

a(n) = d(n, 1) where d(n, k) is defined in A007046. - Sean A. Irvine, Oct 06 2017

The ordinary generating function is f(1), where f(v) satisfies the functional equation f(v) = v*(1 + f(1 + x*v) - f(1)). The variable x marks inversions and v marks left-to-right minima. - David Bevan, Dec 19 2017

EXAMPLE

a(3) = 5; five indecomposable permutations have three inversions: 321, 2341, 2413, 3142, 4123. - David Bevan, Dec 19 2017

MATHEMATICA

r[1, 1]=1; r[_, 0]:=0; r[n_, k_]:=r[n, k]=Sum[r[n-k, j]Binomial[j+1, k], {j, k-1, (Sqrt[8(n-k)+1]-1)/2}]; a[n_]:=Sum[r[n, k], {k, (Sqrt[8n+1]-1)/2}]; Array[a, 20] (* David Bevan, Dec 19 2017 *)

CROSSREFS

Cf. A007046, A007048, A007049, A007050, A007076.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Sean A. Irvine, Oct 06 2017

STATUS

approved

A007046

Number of irreducible positions of size n in Montreal solitaire.
(Formerly M2800)

+10
5

1, 3, 9, 25, 70, 194, 537, 1485, 4104, 11338, 31318, 86498, 238885, 659713, 1821843, 5031071, 13893316, 38366206, 105947374, 292570493, 807923428, 2231050832, 6160961041, 17013250192, 46981405457, 129737238488, 358264064448, 989331456469

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=3..30.

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

FORMULA

a(n) = d(n, 2) where d(n, k) = 0 if n < k*(k+1)/2, d(n, k) = 1 if n = k*(k+1)/2, and d(n, k) = d(n, k+1) + Sum_{r=1..k} binomial(k + 1, r) * d(n - k*(k+1)/2 + r*(r-1)/2, r) if n > k*(k+1)/2 [From Cannings and Haigh]. - Sean A. Irvine, Sep 25 2017

CROSSREFS

Cf. A007048, A007049, A007050, A007075, A007076.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Sean A. Irvine, Sep 25 2017

STATUS

approved

A007048

Number of irreducible positions of size n in Montreal solitaire.
(Formerly M1246)

+10
5

1, 2, 4, 11, 28, 77, 209, 573, 1576, 4340, 11964, 33004, 91080, 251407, 694065, 1916306, 5291223, 14610468, 40344380, 111406090, 307637516, 849517917, 2345890032, 6478044971, 17888793238, 49399054113, 136413247359, 376699178348, 1040238528205

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..29.

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

CROSSREFS

Cf. A007046, A007049, A007050, A007075, A007076.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Sean A. Irvine, Sep 25 2017

STATUS

approved

A007050

Number of irreducible positions of size n in Montreal solitaire.
(Formerly M1754)

+10
5

1, 2, 7, 21, 57, 162, 452, 1255, 3474, 9621, 26604, 73531, 203166, 561242, 1550216, 4281502, 11824338, 32654467, 90177615, 249028277, 687692923, 1899056307, 5244202459, 14481714130, 39990771798, 110433069740, 304956719409, 842126002534

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=6..33.

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

CROSSREFS

Cf. A007046, A007048, A007049, A007075, A007076.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Sean A. Irvine, Sep 25 2017

STATUS

approved

A007076

Number of irreducible positions of size n in Montreal solitaire.
(Formerly M3477)

+10
5

1, 4, 14, 42, 123, 351, 988, 2761, 7682, 21313, 59029, 163314, 451529, 1247842, 3447574, 9523375, 26303825, 72646588, 200627795, 554056162, 1530058222, 4225290890, 11668141901, 32221405479, 88978639438, 245711832196, 678524838258

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=6..32.

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

FORMULA

a(n) = d(n, 3) where d(n, k) is defined in A007046. - Sean A. Irvine, Oct 06 2017

CROSSREFS

Cf. A007046, A007048, A007049, A007050, A007075.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Sean A. Irvine, Oct 06 2017

STATUS

approved

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